UPSC CDS 1 2022 Math Paper

Given :
Number of factors\/divisor of 1000 (excluding 1 and 1000)
Formula used :
For N=xa×yb×zc
Where x,y and z are prime numbers and a,b and c are their powers
Number of factors =(a+1)(b+1)(c+1)
Calculations :
Prime factorization of 1000
⇒1000=(2×5)3
⇒1000=23×53
Hence, number of factors =(3+1)(3+1)
⇒4×4=16
The total number of factors or divisors for 1000 is 16 .
Now excluding 1 and 1000 , we get
⇒16−2
⇒14
∴ The number of divisors of 1000 (excluding 1 and 1000 ) is 14.