Difference between revisions of "2003 AMC 10B Problems/Problem 25"
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− | ==Problem | + | ==Problem== |
How many distinct four-digit numbers are divisible by <math>3</math> and have <math>23</math> as their last two digits? | How many distinct four-digit numbers are divisible by <math>3</math> and have <math>23</math> as their last two digits? | ||
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==Solution== | ==Solution== | ||
− | ===Solution 1=== | + | ===Solution 1 (Slow)=== |
To test if a number is divisible by three, you add up the digits of the number. If the sum is divisible by three, then the original number is a multiple of three. If the sum is too large, you can repeat the process until you can tell whether it is a multiple of three. This is the basis for the solution. Since the last two digits are <math> 23 </math>, the sum of the digits is <math> 2+3 = 5 </math> (therefore it is not divisible by three). However certain numbers can be added to make the sum of the digits a multiple of three. | To test if a number is divisible by three, you add up the digits of the number. If the sum is divisible by three, then the original number is a multiple of three. If the sum is too large, you can repeat the process until you can tell whether it is a multiple of three. This is the basis for the solution. Since the last two digits are <math> 23 </math>, the sum of the digits is <math> 2+3 = 5 </math> (therefore it is not divisible by three). However certain numbers can be added to make the sum of the digits a multiple of three. | ||
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<cmath> 1+4+7+9+6+3 = \boxed{\textbf{(B)}\ 30} </cmath> | <cmath> 1+4+7+9+6+3 = \boxed{\textbf{(B)}\ 30} </cmath> | ||
− | + | ||
− | ===Solution 2=== | + | ===Solution 2 (Medium)=== |
A number divisible by <math>3</math> has all its digits add to a multiple of <math>3.</math> The last two digits are <math>2</math> and <math>3</math> and add up to <math>5 \equiv 2\ (\text{mod}\ 3).</math> Therefore the first two digits must add up to <math>1\ (\text{mod}\ 3).</math> <math>4</math> digits (including <math>0</math>) are <math>0\ (\text{mod}\ 3),</math> <math>3</math> are <math>1\ (\text{mod}\ 3),</math> and <math>3</math> are <math>2\ (\text{mod}\ 3).</math> The following combinations are equivalent to <math>1\ (\text{mod}\ 3)</math>: | A number divisible by <math>3</math> has all its digits add to a multiple of <math>3.</math> The last two digits are <math>2</math> and <math>3</math> and add up to <math>5 \equiv 2\ (\text{mod}\ 3).</math> Therefore the first two digits must add up to <math>1\ (\text{mod}\ 3).</math> <math>4</math> digits (including <math>0</math>) are <math>0\ (\text{mod}\ 3),</math> <math>3</math> are <math>1\ (\text{mod}\ 3),</math> and <math>3</math> are <math>2\ (\text{mod}\ 3).</math> The following combinations are equivalent to <math>1\ (\text{mod}\ 3)</math>: | ||
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<cmath>3\cdot3 + 3\cdot4 + 3\cdot3 = 9 + 12 + 9 = \boxed{\textbf{(B)}\ 30}</cmath> | <cmath>3\cdot3 + 3\cdot4 + 3\cdot3 = 9 + 12 + 9 = \boxed{\textbf{(B)}\ 30}</cmath> | ||
− | ===Solution 3 === | + | ===Solution 3 (Fast)=== |
We have the following: <math> n \equiv 0 \pmod{3}</math> and <math>n \equiv 23\pmod{100}</math>. Then <math>n = 3a = 100b+23</math> for some integers <math>a</math> and <math>b</math>. Taking mod 3 gives: | We have the following: <math> n \equiv 0 \pmod{3}</math> and <math>n \equiv 23\pmod{100}</math>. Then <math>n = 3a = 100b+23</math> for some integers <math>a</math> and <math>b</math>. Taking mod 3 gives: | ||
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<math>999 < 300c+123 < 10000</math> so <math>876 < 300c < 9877</math>. Dividing by <math>300</math> gives <math>2.92 < c < 32.9222</math> so <math>3\le c \le 32</math>. This gives <math>32-3+1 = \boxed{\textbf{(B)} \ 30 }</math> | <math>999 < 300c+123 < 10000</math> so <math>876 < 300c < 9877</math>. Dividing by <math>300</math> gives <math>2.92 < c < 32.9222</math> so <math>3\le c \le 32</math>. This gives <math>32-3+1 = \boxed{\textbf{(B)} \ 30 }</math> | ||
− | ===Solution 4( | + | ===Solution 4 (Extremely Fast)=== |
The number is in the form <math>xy23</math> | The number is in the form <math>xy23</math> | ||
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Hence, there are <math>\frac{97-10}{3}+1=30</math> distinct four digit numbers. | Hence, there are <math>\frac{97-10}{3}+1=30</math> distinct four digit numbers. | ||
− | == Solution 5 (Even Faster) == | + | === Solution 5 (Even Faster than Extremely Fast) === |
+ | |||
+ | Following the form <math>xy23</math> in Solution 4, notice that <math>x+y \equiv 1 \pmod 3</math> to satisfy our condition. Choose a value of <math>x</math> from 1 to 9. For <math>x=1, 4, 7</math>, there are exactly 4 values of <math>y: 0, 3, 6, 9</math>. For the remaining 6 digits, there are 3 choices for y. So our answer is <math>(3)(4)+(6)(3)=30</math>. | ||
+ | |||
+ | === Solution 6 (Even Faster than Even Faster than Extremely Fast)=== | ||
+ | There are <math>9 \cdot 10 = 90</math> possible values for the first two digits. <math>\frac{1}{3}</math> of them yield a multiple of <math>3</math>, so the answer is <math>\frac{90}{3} = \boxed{\textbf{(B)}\ 30}</math> | ||
+ | |||
+ | (In more detail: There are 90 possible values for the first two digits. All numbers can be written as 0, 1, or 2 mod(3). 2+3= 5 = -1 mod(3), so the first two digits need to add to 1 mod(3) for all the digits to equal 0 mod(3) (a multiple of 3). Out of the 3 ways to pick 2 numbers in mod(3) (0+1, 0+2, and 1+2), only <math>\frac{1}{3}</math> of the ways (only 0+1) equals 1 mod(3). Thus, the answer is <math>\frac{90}{3} = \boxed{\textbf{(B)}\ 30}</math>. | ||
+ | |||
+ | === Solution 7 (Easy to understand) === | ||
+ | Denote the four digit number by <math>ab23_{10}</math>. Then it is well known the sum of the digits of an integer <math>N</math> determine whether or not they are divisible by <math>3</math>. Thus we want <math>a + b + 2 + 3 \equiv 0 \pmod 3</math>. | ||
+ | |||
+ | <math>\underline{a = 1}: \implies</math> <math>b \equiv 0 \pmod 3 \implies b = 0, 3, 6, 9</math>. | ||
+ | |||
+ | <math>\underline{a = 2}: \implies</math> <math>b \equiv 2 \pmod 3 \implies b = 2, 5, 8</math>. | ||
+ | |||
+ | <math>\underline{a = 3}: \implies</math> <math>b \equiv 1 \pmod 3 \implies b = 1, 4, 7</math>. | ||
+ | |||
+ | |||
+ | Now we have that when <math>a = 4</math> this is the same case as <math>a = 1</math> as <math>4 \equiv 1 \pmod 3</math>. Hence if we count the above number of solutions and multiply by <math>3</math> (as <math>1 \leq a \leq 9</math>), we are home free. There are <math>4 + 3 + 3 = 10</math> solutions. Multiplying this by <math>3</math> yields <math>\boxed{30}</math>. <math>\blacksquare</math> | ||
− | + | === Solution 8 (Very very inefficient and slow) === | |
+ | Don't do this on a test unless you have lots of time left. | ||
+ | |||
+ | Pounding out all of the possible numbers and counting them, we get <math>\boxed{\textbf{(B)}\ 30}.</math> | ||
== See also == | == See also == | ||
{{AMC10 box|year=2003|num-b=24|after=Last problem|ab=B}} | {{AMC10 box|year=2003|num-b=24|after=Last problem|ab=B}} | ||
+ | |||
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 15:56, 17 August 2024
Contents
Problem
How many distinct four-digit numbers are divisible by and have
as their last two digits?
Solution
Solution 1 (Slow)
To test if a number is divisible by three, you add up the digits of the number. If the sum is divisible by three, then the original number is a multiple of three. If the sum is too large, you can repeat the process until you can tell whether it is a multiple of three. This is the basis for the solution. Since the last two digits are , the sum of the digits is
(therefore it is not divisible by three). However certain numbers can be added to make the sum of the digits a multiple of three.
,
, and so on.
However since the largest four-digit number ending with is
, the maximum sum is
.
Using that process we can fairly quickly compile a list of the sum of the first two digits of the number.
Now we find all the two-digit numbers that have any of the sums shown above. We can do this by listing all the two digit numbers in separate cases.
And finally, we add the number of elements in each set.
Solution 2 (Medium)
A number divisible by has all its digits add to a multiple of
The last two digits are
and
and add up to
Therefore the first two digits must add up to
digits (including
) are
are
and
are
The following combinations are equivalent to
:
Let the first term in each combination represent the thousands digit and the second term represent the hundreds digit. We can use this to find the total amount of four-digit numbers.
Solution 3 (Fast)
We have the following: and
. Then
for some integers
and
. Taking mod 3 gives:
so
for some integer
. But
so
. Bounding this gives us:
so
. Dividing by
gives
so
. This gives
Solution 4 (Extremely Fast)
The number is in the form
Notice that the number is divisible by if the sum of the digits of
is divisible by
Our first number that is divisible by is
, next is
..
Notice
goes from
Hence, there are distinct four digit numbers.
Solution 5 (Even Faster than Extremely Fast)
Following the form in Solution 4, notice that
to satisfy our condition. Choose a value of
from 1 to 9. For
, there are exactly 4 values of
. For the remaining 6 digits, there are 3 choices for y. So our answer is
.
Solution 6 (Even Faster than Even Faster than Extremely Fast)
There are possible values for the first two digits.
of them yield a multiple of
, so the answer is
(In more detail: There are 90 possible values for the first two digits. All numbers can be written as 0, 1, or 2 mod(3). 2+3= 5 = -1 mod(3), so the first two digits need to add to 1 mod(3) for all the digits to equal 0 mod(3) (a multiple of 3). Out of the 3 ways to pick 2 numbers in mod(3) (0+1, 0+2, and 1+2), only of the ways (only 0+1) equals 1 mod(3). Thus, the answer is
.
Solution 7 (Easy to understand)
Denote the four digit number by . Then it is well known the sum of the digits of an integer
determine whether or not they are divisible by
. Thus we want
.
.
.
.
Now we have that when this is the same case as
as
. Hence if we count the above number of solutions and multiply by
(as
), we are home free. There are
solutions. Multiplying this by
yields
.
Solution 8 (Very very inefficient and slow)
Don't do this on a test unless you have lots of time left.
Pounding out all of the possible numbers and counting them, we get
See also
2003 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 24 |
Followed by Last problem | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
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