Other Topics Puzzle – Challenge 97

This is a great mathematics puzzle and brain teaser which contains some mathematical content. Can you solve it? The full solution is also given.

Other Topics Puzzle – Challenge 97


What is the smallest positive integer that has 7 factors?

The Absolute Best Book to Challenge Your Smart Student!

Satisfied 123 Students

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

The Best Books to Ace Algebra

Satisfied 1 Students
Satisfied 92 Students
Satisfied 125 Students

Related to This Article

What people say about "Other Topics Puzzle – Challenge 97 - Effortless Math: We Help Students Learn to LOVE Mathematics"?

No one replied yet.

Leave a Reply

45% OFF

Limited time only!

Save Over 45%

Take It Now!

SAVE $40

It was $89.99 now it is $49.99

The Ultimate Algebra Bundle: From Pre-Algebra to Algebra II