Difference between revisions of "SANSKAR'S OG PROBLEMS"
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<math>02</math> | <math>02</math> | ||
| − | == | + | ==Problem 1== |
| − | Let <math>\overline{ab}</math> be a 2-digit [[positive integer]] satisfying <math>\overline{ab}^2</math> | + | Let <math>\overline{ab}</math> be a 2-digit [[positive integer]] satisfying <math>\overline{ab}^2=a! +b!</math>. Find <math>a+b</math> . |
| + | ==Solution 1 by ddk001 (Casework)== | ||
| + | '''Case 1: <math>a>b</math>''' | ||
| + | In this case, we have | ||
| + | <cmath>\overline{ab}^2=a! +b!=(1+a \cdot (a-1) \cdot \dots \cdot (b+1)) \cdot b! \implies b!|\overline{ab}^2=(10a+b)^2</cmath>. | ||
| + | If | ||
==Problem2 == | ==Problem2 == | ||
For any [[positive integer]] <math>n</math>, <math>n</math>>1 can <math>n!</math> be a [[perfect square]]? If yes, give one such <math>n</math>. If no, then prove it. | For any [[positive integer]] <math>n</math>, <math>n</math>>1 can <math>n!</math> be a [[perfect square]]? If yes, give one such <math>n</math>. If no, then prove it. | ||
Revision as of 22:03, 28 January 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. If you view this page please increment the below number by one:
Problem 1
Let 
be a 2-digit positive integer satisfying 
. Find 
.
Solution 1 by ddk001 (Casework)
Case 1: 
In this case, we have
![\[\overline{ab}^2=a! +b!=(1+a \cdot (a-1) \cdot \dots \cdot (b+1)) \cdot b! \implies b!|\overline{ab}^2=(10a+b)^2\]](http://latex.artofproblemsolving.com/d/6/6/d66bc68298f08f74f02e1d5c7a55f96f1385d389.png)
.
If
Problem2
For any positive integer 
, >1 can 
be a perfect square? If yes, give one such . If no, then prove it.
