UPSC CDS 1 2022 Math Paper

Given :
p=22n+2+m and q=24n−m
n is even natural number
Formula used :
(Mab)=(Ma)b
(Ma+b)=(Ma⋅Mb)
Calculations :
If p is divisible by 5 , we have
⇒p∕5=(22n+2+m)∕5
Using the formula above
⇒((22)n×22+m)∕5
⇒(4n×4+m)∕5
When 4 is divided by 5 it leaves - 1 as a negative remainder, so
⇒(−1n×4+m)∕5
n is an even positive number
⇒(1×4+m)∕5
⇒(4+m)∕5
The least number required for m is 1
⇒(4+1)∕5=5∕5=1
Also, we have q=24n−m
⇒q=16n−m
Since n is an even positive integer. Therefore,
(16-m) will always be one factor of q.
If m=1 then q=16−1=15
Which is divisible by 5 .
∴ The least value of m such that p, as well as q, is divisible by 5 is 1 .