Amazing explanation Sir !!

Regards

Could you please explain how you came up with k=2^2 * 3^2 ? What is the idea behind giving that value to k? Also, the value of k is supposed to be odd. Why did you go with 36 as value of k?

\(k\) represents the no of divisors and which is supposed to be odd.

It should be that \(36\) will give us the smallest \(k\).

By taking \(2^2*3^2\), the \(k\) will be \(3*3 = 9 =\) no of divisor. We can't take odd power else the no of divisors will be even. Not possible in this case.

So when \(k=9\), the positive factors are \(3\) (1,3 and 9) so \(x = 3\)

So these are the smallest values.

Please ask if the doubt remains.

There are three things to note that will make this solution easy.

1. Prime factorisation of a number will allow you to know the number of factors when you multiply the powers of the prime numbers after adding one to each of the powers.
2. Only prime numbers have a total number of factors that is odd.
3. A prime number can only be factored by 1 and itself and hence has two factors.

Going from the above,

A. N and k are prime numbers.
B. Hence if k and x are odd integers greater than 2, the smallest possible value of x is 3.
C. To find k, we have to figure out what perfect square will have only 3 positive factors. This can only be the perfect square of a prime number.
D. When we find k, we have to find the smallest possible number that has k positive factors.

Fo find k, we have to choose the smallest prime number that will be the factor of a perfect square that is still odd.

Let's try 2.

2 squared is 4. 4 is an even number and can't work.
3 squared is 9 which is an odd number so it works.

Hence k is 9

Smallest value of x=3
Smallest value of k=9
Smallest value of N=?

Let's remember that N is also a perfect square.

So let us look at the perfect squares above 9 from the smallest one that will give have 9 factors.

Let us note that since the perfect square will have 9 factors, we need something that meets statement 1 stated above will multiply to form 9.

Since all factors in a prime factorisation will have the lowest power of 1, the only way to achieve 9 is by getting a perfect square with two prime numbers that have powers of 2 giving

(2+1) * (2+1) according to the same statement 1 above.

Let's begin to check the prime factorisations of the perfect squares larger than 9.

4*4 = 16
Prime factorisation: 2 raised to power 4. It has (4+1)=5 factors and hence does not work.
5*5 = 25
Prime factorisation: 5 squared; It has (2+1)=3 factors and hence does not work.
6*6 = 36
Prime factorisation: 2 squared * 3 squared. It has (2+1)*(2+1) =9 factors

Hence the smallest value of N is 36

and N - k - x = 36 - 9 - 3 = 24


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On another note, can someone please teach me how to use the symbols on this platform?