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Contests & Programs AMC and other contests, summer programs, etc.
AMC and other contests, summer programs, etc.
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Contests & Programs AMC and other contests, summer programs, etc.
AMC and other contests, summer programs, etc.
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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
Apr 2, 2025
Spring is in full swing and summer is right around the corner, what are your plans? At AoPS Online our schedule has new classes starting now through July, so be sure to keep your skills sharp and be prepared for the Fall school year! Check out the schedule of upcoming classes below.

WOOT early bird pricing is in effect, don’t miss out! If you took MathWOOT Level 2 last year, no worries, it is all new problems this year! Our Worldwide Online Olympiad Training program is for high school level competitors. AoPS designed these courses to help our top students get the deep focus they need to succeed in their specific competition goals. Check out the details at this link for all our WOOT programs in math, computer science, chemistry, and physics.

Looking for summer camps in math and language arts? Be sure to check out the video-based summer camps offered at the Virtual Campus that are 2- to 4-weeks in duration. There are middle and high school competition math camps as well as Math Beasts camps that review key topics coupled with fun explorations covering areas such as graph theory (Math Beasts Camp 6), cryptography (Math Beasts Camp 7-8), and topology (Math Beasts Camp 8-9)!

Be sure to mark your calendars for the following events:
[list][*]April 3rd (Webinar), 4pm PT/7:00pm ET, Learning with AoPS: Perspectives from a Parent, Math Camp Instructor, and University Professor
[*]April 8th (Math Jam), 4:30pm PT/7:30pm ET, 2025 MATHCOUNTS State Discussion
April 9th (Webinar), 4:00pm PT/7:00pm ET, Learn about Video-based Summer Camps at the Virtual Campus
[*]April 10th (Math Jam), 4:30pm PT/7:30pm ET, 2025 MathILy and MathILy-Er Math Jam: Multibackwards Numbers
[*]April 22nd (Webinar), 4:00pm PT/7:00pm ET, Competitive Programming at AoPS (USACO).[/list]
Our full course list for upcoming classes is below:
All classes run 7:30pm-8:45pm ET/4:30pm - 5:45pm PT unless otherwise noted.

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0 replies
jlacosta
Apr 2, 2025
0 replies
k i Suggestion Form
jwelsh   0
May 6, 2021
Hello!

Given the number of suggestions we’ve been receiving, we’re transitioning to a suggestion form. If you have a suggestion for the AoPS website, please submit the Google Form:
Suggestion Form

To keep all new suggestions together, any new suggestion threads posted will be deleted.

Please remember that if you find a bug outside of FTW! (after refreshing to make sure it’s not a glitch), make sure you’re following the How to write a bug report instructions and using the proper format to report the bug.

Please check the FTW! thread for bugs and post any new ones in the For the Win! and Other Games Support Forum.
0 replies
jwelsh
May 6, 2021
0 replies
k i Read me first / How to write a bug report
slester   3
N May 4, 2019 by LauraZed
Greetings, AoPS users!

If you're reading this post, that means you've come across some kind of bug, error, or misbehavior, which nobody likes! To help us developers solve the problem as quickly as possible, we need enough information to understand what happened. Following these guidelines will help us squash those bugs more effectively.

Before submitting a bug report, please confirm the issue exists in other browsers or other computers if you have access to them.

For a list of many common questions and issues, please see our user created FAQ, Community FAQ, or For the Win! FAQ.

What is a bug?
A bug is a misbehavior that is reproducible. If a refresh makes it go away 100% of the time, then it isn't a bug, but rather a glitch. That's when your browser has some strange file cached, or for some reason doesn't render the page like it should. Please don't report glitches, since we generally cannot fix them. A glitch that happens more than a few times, though, could be an intermittent bug.

If something is wrong in the wiki, you can change it! The AoPS Wiki is user-editable, and it may be defaced from time to time. You can revert these changes yourself, but if you notice a particular user defacing the wiki, please let an admin know.

The subject
The subject line should explain as clearly as possible what went wrong.

Bad: Forum doesn't work
Good: Switching between threads quickly shows blank page.

The report
Use this format to report bugs. Be as specific as possible. If you don't know the answer exactly, give us as much information as you know. Attaching a screenshot is helpful if you can take one.

Summary of the problem:
Page URL:
Steps to reproduce:
1.
2.
3.
...
Expected behavior:
Frequency:
Operating system(s):
Browser(s), including version:
Additional information:


If your computer or tablet is school issued, please indicate this under Additional information.

Example
Summary of the problem: When I click back and forth between two threads in the site support section, the content of the threads no longer show up. (See attached screenshot.)
Page URL: http://artofproblemsolving.com/community/c10_site_support
Steps to reproduce:
1. Go to the Site Support forum.
2. Click on any thread.
3. Click quickly on a different thread.
Expected behavior: To see the second thread.
Frequency: Every time
Operating system: Mac OS X
Browser: Chrome and Firefox
Additional information: Only happens in the Site Support forum. My tablet is school issued, but I have the problem at both school and home.

How to take a screenshot
Mac OS X: If you type ⌘+Shift+4, you'll get a "crosshairs" that lets you take a custom screenshot size. Just click and drag to select the area you want to take a picture of. If you type ⌘+Shift+4+space, you can take a screenshot of a specific window. All screenshots will show up on your desktop.

Windows: Hit the Windows logo key+PrtScn, and a screenshot of your entire screen. Alternatively, you can hit Alt+PrtScn to take a screenshot of the currently selected window. All screenshots are saved to the Pictures → Screenshots folder.

Advanced
If you're a bit more comfortable with how browsers work, you can also show us what happens in the JavaScript console.

In Chrome, type CTRL+Shift+J (Windows, Linux) or ⌘+Option+J (Mac).
In Firefox, type CTRL+Shift+K (Windows, Linux) or ⌘+Option+K (Mac).
In Internet Explorer, it's the F12 key.
In Safari, first enable the Develop menu: Preferences → Advanced, click "Show Develop menu in menu bar." Then either go to Develop → Show Error console or type Option+⌘+C.

It'll look something like this:
IMAGE
3 replies
slester
Apr 9, 2015
LauraZed
May 4, 2019
k i Community Safety
dcouchman   0
Jan 18, 2018
If you find content on the AoPS Community that makes you concerned for a user's health or safety, please alert AoPS Administrators using the report button (Z) or by emailing sheriff@aops.com . You should provide a description of the content and a link in your message. If it's an emergency, call 911 or whatever the local emergency services are in your country.

Please also use those steps to alert us if bullying behavior is being directed at you or another user. Content that is "unlawful, harmful, threatening, abusive, harassing, tortuous, defamatory, vulgar, obscene, libelous, invasive of another's privacy, hateful, or racially, ethnically or otherwise objectionable" (AoPS Terms of Service 5.d) or that otherwise bullies people is not tolerated on AoPS, and accounts that post such content may be terminated or suspended.
0 replies
dcouchman
Jan 18, 2018
0 replies
SMT Online 2025 Certificates/Question Paper/Grading
techb   5
N a few seconds ago by Inaaya
It is May 1st. I have been anticipating the arrival of my results displayed in the awards ceremony in the form of a digital certificate. I have unfortunately not received anything. I have heard from other sources(AoPS, and the internet), that the certificates generally arrive at the end of the month. I would like to ask the organizers, or the coordinators of the tournament, to at least give us an ETA. I would like to further elaborate on the expedition of the release of the Question Papers and the grading. The question papers would be very helpful to the people who have taken the contest, and also to other people who would like to solve them. It would also help, as people can discuss the problems that were given in the test, and know different strategies to solve a problem they have solved. In regards to the grading, it would be a crucial piece of evidence to dispute the score shown in the awards ceremony, in case the contestant is not satisfied.
5 replies
techb
4 hours ago
Inaaya
a few seconds ago
ranttttt
alcumusftwgrind   22
N 2 minutes ago by idk12345678
rant
22 replies
alcumusftwgrind
Yesterday at 11:04 PM
idk12345678
2 minutes ago
Too Bad I'm Lactose Intolerant
hwl0304   218
N an hour ago by Maximilian113
Source: 2018 USAMO Problem 1/USAJMO Problem 2
Let \(a,b,c\) be positive real numbers such that \(a+b+c=4\sqrt[3]{abc}\). Prove that \[2(ab+bc+ca)+4\min(a^2,b^2,c^2)\ge a^2+b^2+c^2.\]
218 replies
hwl0304
Apr 18, 2018
Maximilian113
an hour ago
Please support!
warriorsin7   0
2 hours ago
warriorsin7
2 hours ago
0 replies
k How to delete a private forum you created
Platinum_Dragon   2
N Yesterday at 9:36 PM by jlacosta
Is this possible? thank you
2 replies
Platinum_Dragon
Yesterday at 4:26 PM
jlacosta
Yesterday at 9:36 PM
This has been going on for a while now, can anyone else relate?
jmr2010   2
N Yesterday at 3:02 AM by valisaxieamc
Most of the time when I type in something for the tags or search for a user, the AoPS suggestion box pops up, and most of the time, when I click the suggestion, the box just disappears, meaning the automatic system usually never works
2 replies
jmr2010
Apr 29, 2025
valisaxieamc
Yesterday at 3:02 AM
k How to remove tags from a PM after you've removed yourself from it
Platinum_Dragon   2
N Apr 29, 2025 by Platinum_Dragon
Is it possible? Because it's kind of annoying to have a whole bunch of tags that stick around forever.

thank you
2 replies
Platinum_Dragon
Apr 29, 2025
Platinum_Dragon
Apr 29, 2025
k Reaper....
Happycat2   22
N Apr 29, 2025 by jlacosta
Can someone explain what the reaper is this time? I'm sorry but I don't know what "Rapper ear error a pear perrier ear ape ea games" means.
22 replies
Happycat2
Apr 27, 2025
jlacosta
Apr 29, 2025
k Reaper Description
Jwenslawski   1
N Apr 29, 2025 by LostInBali
What does the Reaper game description mean?

"rapper eaE error a Sear perriTr ear apeIea games"

Is it in another language?
1 reply
Jwenslawski
Apr 29, 2025
LostInBali
Apr 29, 2025
k blog glitch?
ngelyy   18
N Apr 29, 2025 by Blossom_tree_17
RESOLVED
18 replies
ngelyy
Apr 29, 2025
Blossom_tree_17
Apr 29, 2025
k Hieroglyph (and Petroglyph?) on Post #1
awesomeming327.   10
N Apr 28, 2025 by bpan2021
Steps to Reproduce:
1. Click this link
2. Scroll up.
Behavior: it should show two copies of a certain "post #0" above post #1.
Frequency: 100%

Mac OS, Arc Browser (which runs on Chromium)

Note: you can also click on "view topic" on the N7 of the 2023 ISL
10 replies
awesomeming327.
Apr 27, 2025
bpan2021
Apr 28, 2025
k something glitched?
JohannIsBach   10
N Apr 27, 2025 by valisaxieamc
hi. my aops whenever i go 2 the website the website is like way to big and i cant access anything? i am on via an ingognito window but its kind of inconvenient not sure if its just my laptop or the website. i cant recreat it on any other device? plz help [/resolved]
10 replies
JohannIsBach
Apr 25, 2025
valisaxieamc
Apr 27, 2025
k alcumus not working? [RESOLVED]
JohannIsBach   3
N Apr 27, 2025 by MathDolphin95
4 some reason alcumus isnt working??? in the 1st attachment there isnt any "play, profile, ...etc."
and in the second pic i clicked the "play" button, but its just a blank screen??? i tried refreshing it didnt work????
3 replies
JohannIsBach
Apr 27, 2025
MathDolphin95
Apr 27, 2025
Is this just my device? Typing glitch
pearlhaas   4
N Apr 26, 2025 by valisaxieamc
This started today.
When I tried to type in the AoPS homework section (as well as forums), where you put your answers into the slots, I found that I couldn’t? This might only be my device but it was delaying my work and I just thought I’d share.
I refreshed multiple times, but it only works sometimes.
Sometimes, the blinking blue bar shows but I can’t type, and other times, the blue bar doesn’t show at all.
I am able to type this forum up since this was one of the few times that my device let me type (for more, I have an iPad. Not sure what generation, but it’s a pretty recent version, I believe).
If this is a problem with my device, I apologize for wasting your time :)
Just wanted to make sure I could get this notice out if others have the same problem.
4 replies
pearlhaas
Apr 25, 2025
valisaxieamc
Apr 26, 2025
basic nt
zhoujef000   39
N Apr 20, 2025 by NicoN9
Source: 2025 AIME I #1
Find the sum of all integer bases $b>9$ for which $17_b$ is a divisor of $97_b.$
39 replies
zhoujef000
Feb 7, 2025
NicoN9
Apr 20, 2025
basic nt
G H J
G H BBookmark kLocked kLocked NReply
Source: 2025 AIME I #1
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zhoujef000
311 posts
#1 • 2 Y
Y by PikaPika999, Soupboy0
Find the sum of all integer bases $b>9$ for which $17_b$ is a divisor of $97_b.$
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chessboy_123
168 posts
#2 • 1 Y
Y by PikaPika999
21 and 49 --> 070?
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plang2008
337 posts
#3 • 1 Y
Y by PikaPika999
$b + 7 \mid 9b + 7 \implies b + 7 \mid -56$ so $b = 21, 49 \implies \boxed{070}$.
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QuelFromage
14 posts
#4
Y by
$9b+7 \equiv 0~(\text{mod }b+7) \implies 9b+7 - 9(b+7) \equiv 0~(\text{mod } b+7) \implies -56 \equiv 0~(\text{mod } b+7) \implies b = 21, 49 \implies \boxed{070}$
Z K Y
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MathPerson12321
3745 posts
#5
Y by
QuelFromage wrote:
$9b+7 \equiv 0~(\text{mod }b+7) \implies 9b+7 - 9(b+7) \equiv 0~(\text{mod } b+7) \implies -56 \equiv 0~(\text{mod } b+7) \implies b = 21, 49 \implies \boxed{070}$

I forgot how to modulo so I spent a ton of time looking for other sols until I realized it was just 70.
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ChaitraliKA
1004 posts
#6
Y by
I did Simon's factoring trick lol
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fruitmonster97
2489 posts
#7
Y by
$b+7|9b+7$ so $b+7|8b$ so let $k(b+7)=8b$ and check $k<8$ to find $21+49=\boxed{070}$
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xHypotenuse
776 posts
#8
Y by
Let a = b+7 and then you find that b+7 | 56 so only b = 21 and b= 49 work when b>9. Therefore, 21 + 40 = 070.
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darkgrimoire
5 posts
#9
Y by
Did anyone use polynomial division to get 9-\frac{56}{b+7}. Then b+7=28, and b+7 =56. I forgot all the rules of divisibility lol.
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BS2012
1026 posts
#10 • 1 Y
Y by darkgrimoire
darkgrimoire wrote:
Did anyone use polynomial division to get 9-\frac{56}{b+7}. Then b+7=28, and b+7 =56. I forgot all the rules of divisibility lol.

yea thats what i did
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darkgrimoire
5 posts
#11
Y by
darkgrimoire wrote:
Did anyone use polynomial division to get $9-\frac{56}{b+7}$. Then b+7=28, and b+7 =56. I forgot all the rules of divisibility lol.
Z K Y
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DreamineYT
285 posts
#12
Y by
Click to reveal hidden text
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megahertz13
3183 posts
#13
Y by
Notice that we have $$7b+9\equiv 0\pmod {b+9}\implies 56\equiv 0\pmod {b+9}.$$Now by inspection, $b=21$ and $b=49$ are the only solutions. This yields the answer $\boxed{070}$.
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sixoneeight
1138 posts
#14
Y by
Easy things give $b+7|56$. Thus, the possible values greater than $9$ are $14, 28, 56$.

1 hour later

Oh wait, forgot to subtract $7$ hahaha
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megarnie
5603 posts
#15 • 1 Y
Y by Sedro
Fakesolve that gives right answer:

$b + 7 \mid 9b + 7\implies b + 7 \mid 9b$ (this is wrong but it's what i did), so $b + 7 \mid 63$. Thus, $b \in \{14,56\}$, so the answer is $\boxed{070}$.
This post has been edited 3 times. Last edited by megarnie, Feb 7, 2025, 5:52 PM
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MathRook7817
666 posts
#16
Y by
megarnie wrote:
Fakesolve that gives right answer:

$b + 7 \mid 9b + 7\implies b + 7 \mid 9b$ (this is wrong but it's what i did), so $b + 7 \mid 63$. Thus, $b \in \{14,56\}$, so the answer is $\boxed{070}$.

so lucky lol
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AshAuktober
1000 posts
#17
Y by
$b+7\mid 9b+7\iff b+7 \mid 56$, so $b = 21, 49$.Therefore the answer is $\boxed{70}$.
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theorz
6 posts
#18
Y by
ez it is 70
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pingpongmerrily
3581 posts
#19
Y by
b+7 divides 9b+7
b-49 divides b+7
b+7 divides 56
b=21, b=49
070
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lpieleanu
2980 posts
#20
Y by
Solution
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junlongsun
70 posts
#21
Y by
$\frac{9b+7}{b+7}$ has to be an integer
Notice that we can rewrite $9b+7$ as $9(b+7)-56$
$$\frac{9b+7}{b+7}=\frac{9(b+7)-56}{b+7}=\frac{9(b+7)}{b+7}+\frac{-56}{b+7}$$Because $\frac{9(b+7)}{b+7}=9$, we just need to make sure $\frac{-56}{b+7}$ is an integer.

The only factors of 56 that satisfy $b>9$, $b+7=28$, $b+7=56$
$$b=21, b=49$$$$49+21=70$$$$\fbox{70}$$
This post has been edited 3 times. Last edited by junlongsun, Feb 8, 2025, 12:05 AM
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apple143
62 posts
#22
Y by
AshAuktober wrote:
$b+7\mid 9b+7\iff b+7 \mid 56$, so $b = 21, 49$.Therefore the answer is $\boxed{70}$.

yeah this is what i got. wrote almost nothing down
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ashays
91 posts
#23
Y by
9b+7 is divisible by b+7, so 9b+7 - 9(b+7) will still be divisible by b+7, so -56 is divisible by b+7. we first try -56-7=-49, so one b possibility could be 49. 49-7=42, 42/2 = 21, 21 is another possibility. 21-7=14, 14/2 = 7, which is less than 9, so we only have 21 and 49, giving us 70.
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RedFireTruck
4221 posts
#24
Y by
We want $(b+7)|(9b+7)$ so $(b+7)|56$ so the answer is $(56-7)+(28-7)=49+21=\boxed{070}$.
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gracemoon124
872 posts
#25
Y by
$17_b$ dividing $97_b$ means that $b+7\mid 9b+7$, and note that $9b+7=9(b+7)-56$, so $b+7\mid 56$. Then $b$ could be $21$ or $49$ which means $\boxed{070}$.

remarks: while mocking this i had a worse solution (bounding based on value of $\tfrac{9b+7}{b+7}$ WHICH WORKS THOUGH) and it takes about the same amount of time (:
This post has been edited 1 time. Last edited by gracemoon124, Feb 8, 2025, 6:17 PM
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pog
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#26
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if you do not know any number theory:

By the definition of a divisor, $(b+7)k = 9b+7$ for an integer $k$. Clearly $k \ge 9$ does not work. So now we can just try every single value.


$b+7=9b+7$ no
$2b+14=9b+7$ gives $7b=7$ which doesn't work
$3b+21=9b+7$ gives $6b=14$ which doesn't work
$4b+28=9b+7$ gives $5b=21$ which doesn't work
$5b+35=9b+7$ gives $4b=28$ which doesn't work
$6b+42=9b+7$ gives $3b=35$ which doesn't work
$7b+49=9b+7$ gives $2b=42$ which does work
$8b+56=9b+7$ gives $1b=49$ which does work


Hence our answer is $21+49=\boxed{70}$.
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Mr.Sharkman
498 posts
#27
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LMFAO this is what I thought p1 was gonna be: What is the sum of all $n$ for which $15_{n}$ divides $2025_{n}$?
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pingpongmerrily
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#28
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Mr.Sharkman wrote:
LMFAO this is what I thought p1 was gonna be: What is the sum of all $n$ for which $15_{n}$ divides $2025_{n}$?

that might be hard
uh (n+5) divides $2n^3+2n+5$
is that even factorable
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sunannie
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#29
Y by
pog wrote:
if you do not know any number theory:

By the definition of a divisor, $(b+7)k = 9b+7$ for an integer $k$. Clearly $k \ge 9$ does not work. So now we can just try every single value.


$b+7=9b+7$ no
$2b+14=9b+7$ gives $7b=7$ which doesn't work
$3b+21=9b+7$ gives $6b=14$ which doesn't work
$4b+28=9b+7$ gives $5b=21$ which doesn't work
$5b+35=9b+7$ gives $4b=28$ which doesn't work
$6b+42=9b+7$ gives $3b=35$ which doesn't work
$7b+49=9b+7$ gives $2b=42$ which does work
$8b+56=9b+7$ gives $1b=49$ which does work


Hence our answer is $21+49=\boxed{70}$.

thats what i did too
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xTimmyG
265 posts
#30 • 1 Y
Y by MrMustache
by the fundamental theory of number theory, the answer must start with an even number. then, guess and check values of b, to get 21+49=70
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MrMustache
2999 posts
#31
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+1 for guess and check its really the only good way to solve this problem. I did all values up to 187 just to be sure.
This post has been edited 1 time. Last edited by MrMustache, Feb 8, 2025, 10:13 PM
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aidan0626
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#32
Y by
Mr.Sharkman wrote:
LMFAO this is what I thought p1 was gonna be: What is the sum of all $n$ for which $15_{n}$ divides $2025_{n}$?

that might be hard
uh (n+5) divides $2n^3+2n+5$
is that even factorable

that wouldn't be that much harder
you get $n+5|255$, and it's basically the same thing
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sansgankrsngupta
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#34
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OG!
$$(17)_b \mid 97_b \iff b^2+7b \mid 9b^2+7b \iff b+7 \mid 8b \iff b+7 \mid 56$$.
Since $b>9$, the only possible $b$ are $b=21,49$ which satisfy.
Thus, the sum of all possible values of $b = 21+49= \boxed{070}$
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sadas123
1251 posts
#35
Y by
xHypotenuse wrote:
Let a = b+7 and then you find that b+7 | 56 so only b = 21 and b= 49 work when b>9. Therefore, 21 + 40 = 070.

what 21+40=61 I think you meant 49??
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sadas123
1251 posts
#36
Y by
I was stuck on this problem for some reason I don't know why

The best way to tackle this problem is to convert everything to variables so first 17 base b = b + 7 and 97 base b = 9b+7 so we can just make a variable when multiplied by it it equals 97 base b

So first to do that we can simplify it like 9b+7 = x(b+7) for some value of x then when we multiply that out we get 9b+7=bx+7x. Because we want to solve for xb we have to subtract that on both sides to get 9b+7-bx= 7x then in any situation like this we have to factor out the b, but first we can subtract 7 on both sides.

When we do that we get b(9-x)= 7(x-1) which when we divide both sides we get

b= (7(x-1))/(9-x) we get this easy equation to solve because we know that x has to be a single digit number because anything greater than 9 won't work so we get that after guess and check x= 7 and 8 so when we plug that in we get 21 and 49, so when we add those together you get $49+21$ $=$ $70$
This post has been edited 2 times. Last edited by sadas123, Feb 9, 2025, 4:44 PM
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Mr.Sharkman
498 posts
#37
Y by
pingpongmerrily wrote:
Mr.Sharkman wrote:
LMFAO this is what I thought p1 was gonna be: What is the sum of all $n$ for which $15_{n}$ divides $2025_{n}$?

that might be hard
uh (n+5) divides $2n^3+2n+5$
is that even factorable

Bruh factor theorem
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A04572
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#38
Y by
b+7 is divisible by 9b+7, but because b+7 is divisible by 9b+63, then b+7 is divisible by 56. And B+7 is greater than 7, so b=21 or 49[Click][sounds stupid but I got b+7 is divisible by 8b, and did a lot of stupid stuff to get -6+-5+-3+0+1+7+21+49=64 cause I forgot b is greater than 9, but luckily I realized this at the end.]
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jasperE3
11281 posts
#39
Y by
zhoujef000 wrote:
Find the sum of all integer bases $b>9$ for which $17_b$ is a divisor of $97_b.$

We need $b+7\mid9b+7$, so $b+7\mid8b$. Let $8b=k(b+7)$, do casework and bounding.
If $k\le4$ then $8b\le4(b+7)$ so $b\le7$, doesn't work.
If $k=5$ then $8b=5b+35$ so $b=\frac{35}3\notin\mathbb Z$, doesn't work.
If $k=6$ then $8b=6b+42$ so $b=21$, which does work.
If $k=7$ then $8b=7b+49$ so $b=49$, which does work.
If $k\ge8$ then $8b\ge8(b+7)$ so $0\ge56$, doesn't work.
Our answer is $21+49=\boxed{070}$.
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Apple_maths60
26 posts
#40
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(97)b =9b+7
(17)b=b+7
So b+7 divides 9b+7
gcd(9b+7,b+7)=b+7
gcd(-56,b+7)=b+7
So we get b+7 divides -56
Now , 56=2*2*2*7 and factors greater than 9+7 are 28 and 56 itself
So, b=21 or 49
Sum =70(answer)
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NicoN9
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#41
Y by
$b+7\mid 9b+7 \Longleftrightarrow b+7\mid -56$ and we easily get $b=21, 49$.
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