Number properties

And another one:

My thoughts on Q2

n=multiple of 5 (or i=n/5)
n=(p^2)*(q)

Since (p^2)*(q) is a multiple of 5, then 5 must also be a factor of (p^2)*(q).

Therefore either p, q or both could be factors of 5 (but we don't know which ones)

To find which also must be factor of 25.

A. p^2 - not necessary because q could be the factor 5 and p could be some other prime number

B. q^2 - not necessary because p could be the factor 5 and q can be some other prime number

C. pq - not necessary because they may not both be factor 5 (and therefore not multiple of 25)

D. p^2*q^2, this works because either p or q (or both) will be the factor 5, so the square will satisfy the requirement of being a multiple of 25.

E. p^3*q -- if q is the factor 5 then this will not necessarily be a multiple of 25

a) Considering the first option that n is not divisible by 2.

N may have values: 3 5 7 9 11 13 15
And their respective
remainders when
n^2 - 1 divided by
24: : 8 0 0 8 0 0 8 ..... and it repeats in this sequence.

So r may be either 0 or 8. hence insufficient.

Similarly (b) is insufficient as it produces different r's.

When (a) and (b) are taken together,

n may have values like 5,7,11,13,17 ... for which when (n^2-1) / 24 will always yield no remainder. So r is 0.

Hence choice (c).

Thanks a lot!
Do you have any suggestions how to improve number properties skills?
That's my only weakness in Q section.