Question

Challenge What is the least whole number that has exactly 7 factors, including 1 and itself? Explain your answer.

Short answer

64 is the least whole number with exactly 7 factors including 1 and itself.

Step-by-step solution

Step-1  . Apply the concept of Variable

In any expression or equation, the variable is an unknown quantity. It may be any alphabet or any special symbol.

Step-2  . Concept used

Perfect squares always have an odd number of factors whereas non perfect squares always have an even number of factors.

$n=p_1^{a1}·p_2^{a2}\mathrm{...}{p}_r^{a_r}$

Step-3  . Find the expression

The first thing you can say just by looking at the question is that the answer will be a perfect square. Perfect squares always have an odd number of factors whereas non perfect squares always have an even number of factors.

There is a formula that gives you the number of factors of a positive integer > 1 from its prime factorization. If you aren’t familiar with the formula, you can always try the perfect squares$2^2,3^2,\dots ,2^2,3^2$,…,until you come to the right one. It will be$8^2=64$$2^6=64.$

The factors of 64 are, 1, 2, 4, 8, 16, 32, and 64.

So, 64 is the least whole number with exactly 7 factors including 1 and itself.