Question

The positive integer k has exactly two positive prime factors, 3 and 7. If k has a total of 6 positive factors, including 1 and k, what is the value of k ? (1) \small 3^{2} is a factor of K. (2) \small 7^{2} is not a factor of K.

Answer 1

Answer:

k = 63

Step-by-step explanation:

Given: k's factors are 1, 3, 7, ___, ___, and k.

(1) $3^{2}$ is a factor of k.

This implies that there are two factors of 3 i..e, 9 is also a factor of k.

Also, $3\times 7=21$ is a factor of k.

Therefore, k's factors are 1, 3, 7, 9, 21, and k.

As there are two 3's and a 7 in factors of k, then is also a factor.

(2) $7^{2}$ is not a factor of K.

If $7^2$ is not a factor of k and total number of factors is 6, there must be two factors of 3 ('K' has exactly two positive prime factors)  as otherwise, if we use a non-prime factor, then k would have more than 6 factors.

So, K's factors are 1, 3, 7, 9, 21, 63.