Chapter 4.1, Problem 70E

To determine

To find the smallest number with 7 factors in total.

Answer to Problem 70E

64

Explanation of Solution

Given Information: Asked to find number with 7 factors including 1 and itself.

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

Calculation:

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  $8^2=64$ $2^6=64.$