Question

http://prnt.sc/b05d8i

Answer 1

@ParthKohli

Answer 2

@rsadhvika Can you upload it to here? That link is not pulling up a picture.

Answer 3

@nfcfox

Answer 4

@ParthKohli

Answer 5

@ganeshie8

Answer 6

.

Answer 7

I'm pretty sure the answer is : \(2^{\text{number of different primes in factorization of \(n\)}-2}\). I made a little program and observed this..

Example:\( 10000 = 100^2 = (2^2 5^2)^2\). Only numbers \(a\) such that \(a(a-1)\) divides 10000 are 625 and 9376. Formula: \(2^{2-1} = 2\). yay...

In general,the number \(a\) must be of the form \(p_1^{k_1}p_2^{k_2}\dotsb p_l^{k_l}\) and \(a-1\) has another decomposition. Since the divisors of \(a-1\) don't divide \(a\), that means all primes \(p_j\) from the decomposition of \(n^2\) must go either in \(a\) or in \(a-1\).

If \(n\) is prime, there are no such numbers \(a\).

Answer 8

EDIT: No, \(a\) is not of the form \(p_1^{k_1}\dotsb p_l^{k_l}\).

Answer 9

But it seems related to the fact that if, for instance, \(n^2 = 2^4 5^2\), \(a\) must have all the factors \(2\) : so \(2^4\). and the \(5^2\) goes to \(a-1\).

Answer 10

Typo. Formula is \(2^{\text{number of different primes in prime factorization of \(n\)}} -2\).

Answer 11

case 1 :
\(a=2^4x\)
\(a-1=5^4y\)

case 2 :
\(a-1=2^4x\)
\(a=5^4y\)

Answer 12

Interesting.. @reemii

Answer 13

@praxer

Answer 14

The formula comes from the fact that all \(a\) are formed by: choosing \(k\) (\(1\le k\le N\)=number of different primes in factorization) primes to go into \(a\). There are \(\binom Nk\) ways to do this. and we can't take \(k=0\) or \(k=N\) because that would mean that \(a-1\) or \(a\) is greater than \(n\).. The sum of the binomial coefficients is \(2^N-1-1\).

Answer 15

For example, for \(n=210=2\times 3 \times 5\times 7\),
a = 1225, factors involved: [7,5]
a = 6076, factors involved: [7,31,2]
a = 9801, factors involved: [3,11]
a = 11025, factors involved: [7,3,5]
a = 15876, factors involved: [7,2,3]
a = 17101, factors involved: [7,349]
a = 17200, factors involved: [43,2,5]
a = 26901, factors involved: [7,3,61]
a = 27000, factors involved: [2,3,5]
a = 28225, factors involved: [5,1129]
a = 33076, factors involved: [2,8269]
a = 34300, factors involved: [7,2,5]
a = 38025, factors involved: [13,3,5]
a = 42876, factors involved: [397,2,3]

Removing the factors that are not 2,3,5,7, we can see the "selection of k out of n"..