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Bunuel
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If the product of all the factors of a positive integer, N 22 Apr 2024, 03:34
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rohithmtr
But then, by default the answer of this question should be 1 right?­
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If the product of all the factors of a positive integer, N 22 Apr 2024, 03:53
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Bunuel
rohithmtr
But then, by default the answer of this question should be 1 right?­
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Ok.. Thank you👍

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If the product of all the factors of a positive integer, N 21 Jun 2026, 20:41
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KarishmaB - those links are not available, do you any other?

plus i can understand that product of factors of N = N^f/2
but i am not getting the reverse of it? could you please explain this in detail?

also, i tried this and got the same answer not sure if this is right

the number of ways of expressing N as a product of 2 coprime = 2^(x-1)
2 primes are there, so 2^2-1 = 2;

Thanks in advance,
Swetha
KarishmaB
This question (which is in two parts) tests your fundamentals of a very interesting topic - Factors

Q.i. If the product of all the factors of a positive integer, N, is \(2^{18}*3^{12}\), how many values can N take?
(A) None
(B) 1
(C) 2
(D) 3
(E) 4

Q.ii. If the product of all the factors of a positive integer, N, is \(2^{9}*3^{9}\), how many values can N take?
(A) None
(B) 1
(C) 2
(D) 3
(E) 4
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If the product of all the factors of a positive integer, N 22 Jun 2026, 09:10
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Hi SwethaReddyL,

You've got the most important piece already: product of factors = N^(f/2). The reverse is just that same equation read in the other direction, plus one consistency check. Let me walk it with Q.i, the way Karishma's (now-broken) links were trying to.

Reversing the formula

Since the product is 2^18 · 3^12 (only the primes 2 and 3), N itself must be built from only those primes:

N = 2^a · 3^b, and its number of factors is f = (a+1)(b+1).

Now plug N into the formula:

N^(f/2) = (2^a · 3^b)^(f/2) = 2^(a·f/2) · 3^(b·f/2)

Match this to the given 2^18 · 3^12, exponent by exponent:

- a · (f/2) = 18
- b · (f/2) = 12

The trick is that f isn't free - it must also equal (a+1)(b+1). So you test a value of f/2 and check both conditions:

- Try f/2 = 6 (so f = 12): then a = 18/6 = 3, b = 12/6 = 2.
- Check: (a+1)(b+1) = 4 · 3 = 12 = f. ✓ It holds, so N = 2^3·3^2 = 72.

No other f/2 makes both the powers integers and the factor-count match - that's why exactly one N works, giving answer B.

Lock the reverse step in with a tiny case

Suppose product of factors = 2^3. Then N = 2^a, and 2^(a(a+1)/2) = 2^3, so a(a+1)/2 = 3 - a = 2 - N = 4. Check: factors of 4 are 1, 2, 4; product = 8 = 2^3. ✓ Same machinery, one prime.

About your 2^(x-1) idea

That formula counts something different - the number of ways to split N into two coprime factors, not how many values of N produce a given product of factors. The two aren't connected, and notice it gives 2^(2-1) = 2, while the answer here is 1. So it lands on the wrong count; it isn't the tool for this question. Stick with matching exponents and checking f = (a+1)(b+1).

Answer: B

SwethaReddyL
KarishmaB - those links are not available, do you any other?

plus i can understand that product of factors of N = N^f/2
but i am not getting the reverse of it? could you please explain this in detail?

also, i tried this and got the same answer not sure if this is right

the number of ways of expressing N as a product of 2 coprime = 2^(x-1)
2 primes are there, so 2^2-1 = 2;

Thanks in advance,
Swetha

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If the product of all the factors of a positive integer, N 23 Jun 2026, 00:51
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SwethaReddyL
KarishmaB - those links are not available, do you any other?

plus i can understand that product of factors of N = N^f/2
but i am not getting the reverse of it? could you please explain this in detail?

also, i tried this and got the same answer not sure if this is right

the number of ways of expressing N as a product of 2 coprime = 2^(x-1)
2 primes are there, so 2^2-1 = 2;

Thanks in advance,
Swetha

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Here are some Factors resources:

Factors Concept Video: https://youtu.be/DxIH8rjhpKY
https://anaprep.com/number-properties-factors-of-a-number/
https://anaprep.com/number-properties-really-knowing-the-factors/
https://anaprep.com/number-properties-factors-of-a-perfect-square/
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Bunuel
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