Question

HOW MANY NUMBERS ARE THERE FROM 1 TO 400 HAVING EXACTLY THREE INTEGRAL FACTORS?; HOW MANY NUMBERS ARE THERE FROM 1 TO 400 HAVING EXACTLY THREE INTEGRAL FACTORS?

Answer 1

Every number has at least 2 factors (except for 1), it can be divided by 1 to yield itself or it can be divided by itself to yield one.

The numbers that only have 2 factors are what we call prime numbers.

So to find a number with 3 factors we need to find a number that has just one more factor.

But when we try other numbers like 6, we see that it has another 2 factors (3 and 2).

Instead we need to find a number that uses the same factor twice.

9 for example is 3x3 using the factor of 3 twice. Hence it only has 3 factors!(9,1,3)

What we notice is that it is all square numbers, with one exception that it's square root must be prime.

I'll leave it up to u to figure out why the second statement is true.