Hi!

(a) To find highest common factor (HCF), you want to multiply the prime factors which are common to the numbers.

180 = 2^2 × 3^2 × 5
784 = 2^4 × 7^2

The largest factors shared by both numbers is 2^2.(since 180 cannot be divided by 7, and 784 cannot be divided by 3 or 5)

So the HCF is 2^2 = 4.


(b) Given the expressions of 180 and 784 as prime factors (shown above), we may express 180x784 as...

180 × 784 = 2^6 × 3^2 × 5 × 7^2

In order for 180×784×m to be cube-rootable, all their prime factors need to be in powers of 3 (or multiples of 3).

So 180 × 784 × m would be 2^6 × 3^3 × 5^3 × 7^3,
m = 3 × 5^2 × 7 = 525


(c)
180 = 2^2 × 3^2 × 5
16 = 2^4

Lowest common multiple (LCM) = 2^4 × 3^2 × 5
= 720

Hence,
180n = 720
n = 4

Hope this helps!