Difference between revisions of "SANSKAR'S OG PROBLEMS"
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Revision as of 04:31, 17 February 2024
Hi, this page is created by ...~ SANSGANKRSNGUPTA This page contains exclusive problems made by me myself. I am the creator of these OG problems. What OG stands for is a secret! Please post your solutions with your name. I will release the Official solutions(can be one or more) by me after the problem has been solved. If you view this page please increment the below number by one:
Contents
Problem 1
Let be a 2-digit positive integer satisfying
. Find
.
Solution 1 (Casework)
Case 1:
In this case, we have
.
If , we must have
, but this contradicts the original assumption of , so hence we must have
.
With this in mind, we consider the unit digit of .
Subcase 1.1:
In this case, we have that
.
There is no apparent contradiction here, so we leave this as it is.
Subcase 1.2:
In this case, we have that
.
This contradicts with the fact that , so this is impossible.
Subcase 1.3:
In this case, we have that
.
However, this is impossible for all .
Subcase 1.4:
In this case, we have that
.
Again, this yields , which, again, contradicts
.
Hence, we must have .
Now, with determined by modular arithmetic, we actually plug in the values.
To simplify future calculations, note that
.
For , this does not hold.
For , this does not hold.
For , this does hold. Hence,
is a solution.
For , this does not hold.
For , this does not hold.
Hence, there is no positive integers and
between
and
inclusive such that
.
Case 2:
For this case, we must have
which is impossible if a is a integer and .
Case 3:
In this case, we have
.
If , we must have
which is impossible since and
.
Hence, .
Subcase 3.1:
Testing cases, we can see that there is no such .
Subcase 3.2:
Testing cases, we can see that there is no such .
Subcase 3.3:
Testing cases, we can see that there is no such .
Subcase 3.4:
Testing cases, we can see that there is no such .
We see that .
~Ddk001
Solution 2 (Official)
cleary square of a positive double-digit integer is either 3-digit or 4-digit as it ranges between
and
.
This means both and
are less than or equal to 7 as any
greater than 7! exceeds 9999.
Now at least one of or
must be greater than or equal to 5 or else the maximum possible value of LHS is 4!+4!=48<100
also, both
and
can't be greater than or equal to 5 as if they were so. The unit digit of LHS would be zero but the i=unit digit of RHS would be either 5,6 or 9.
Hence, one of and
is greater than or equal to 5 and the other is less than 5. this means the unit digit of RHS is the unit digit of a factorial which is less than 5.
Hence unit digit of RHS is 0,1,2, 6 or 4. 0,2,4 and 6 are rejected as follows:-
1. 2 can't be the unit digit of a perfect square.
2. If 6 is the unit digit of OF LHS this means one of the numbers is 3( as only 3! has the unit digit 6) and also this would mean that is either 4 or 6. This directly means that 36 and 34 are the only cases to be tested. 34 can't be as both the digits are less than 5 and 36 clearly doesn't satisfy
3. If 0 is the unit digit of LHS then 50 60 70 are the only cases (as one of the digits is greater than or equal to 5) that don't satisfy
4. If 4 is the unit digit of LHS this means one of the numbers is 4( as only 4! has the unit digit 4) and also this would mean that is either 2 or 8. This directly means that 42 and 48 are the only cases to be tested. 42 can't be as both the digits are less than 5 and 48 can't also as one of the digits is 8
Hence, the unit digit of LHS must be 1 for which =1 or 9(rejected). This means 51 61 and 71 are the only cases to be tried now
which is really very easy to calculate and get 71 as the only possible solution to the problem and
thus, our answer is 7+1=.~ SANSGANKRSNGUPTA
Solution 3 (Official and Fastest)
cleary square of a positive double-digit integer is either 3-digit or 4-digit as it ranges between
and
.
This means both and
are less than or equal to 7 as any
greater than 7! exceeds 9999.
Now at least one of or
must be greater than or equal to 5 or else the maximum possible value of LHS is 4!+4!=48<100
also, both
and
can't be greater than or equal to 5 as if they were so. The unit digit of LHS would be zero but the i=unit digit of RHS would be either 5,6 or 9.
Hence, one of and
is greater than or equal to 5 and the other is less than 5. Also, RHS is a perfect square, so it must be 0 or 1
Hence the number less than 5 can be either 1 or 4 because 2! and 3! are 2
If it is 4, then the unit digit of RHS is 4 meaning is either 2 or 8 both of which aren't possible as 8>7 and the number less than 5 is 4 not 2.
If it is 1 then the unit digit of RHS is 1 implying is either 9(rejected 9>7) or 1. This means
is 1
This means 51 61 and 71 are the only cases to be tried now
which is very easy to calculate and get 71 as the only possible solution to the problem and
thus, our answer is 7+1=.~ SANSGANKRSNGUPTA
Problem2
For any positive integer ,
>1 can
be a perfect square? If yes, give one such
. If no, then prove it.
Problem 3
Let be a positive integer in
10 such that the last 3 digits of any positive integral power of
are the same.
Find the sum of all possible values of the last 3 digits of .
Problem 4
Find all natural numbers such that their square is the sum of the factorials of their digits. (Advanced version of Problem 1)