Other Topics Puzzle – Challenge 97

Challenge:

What is the smallest positive integer that has 7 factors?

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To find the answer, we can factorize numbers from 1, one by one. But, it takes for ever!
Every integer N is the product of powers of prime numbers:
N (= P^{a}Q^{b}…R^{y})
Where P, Q, …,R are prime numbers and a, b, …, y are positive integers.
If N is a power of a prime, then N (= p^{α}), therefore, it has α + 1 factors.
If N (= P^{a}Q^{b}…R^{y}), then, N has (a+1) (b+1) … (y+1) factors.
To find the smallest number that has 7 factors, first write the factors of seven: 7 = 1 × 7
It means that the number in this question has just one prime factor in its decomposition – one with the exponent of α = 6. Keep in mind that b = 0, and (Q^{b} = Q^{0} = 1)
N (= P^{6}Q^{0}). To make N as small as possible, we have to choose the smallest available prime 2. The answer is obviously (N = 2^{6} = 64).
The seven factors of 64 are: 1, 2, 4, 8, 16, 32 and 64

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