This is a great mathematics puzzle and brain teaser which contain some mathematical content. Can you solve it? The full solution is also given.
Challenge:
What is the smallest positive integer that has 7 factors?
The Absolute Best Book to challenge your Smart Student!
The correct answer is 64.
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
