UPSC CAPF Assistant Commandant Model Paper 5

GIVEN:
[517×1415×168×1510×124×2522×1014×24p] is divisible by 1082
CONCEPT:
10 = 2 × 5
Whenever the expression is divisible by 10, there must have a pair of ‘2 and 5’ in the expression.
So we need to count the total possible pairs of ‘2 and 5’ in the expression to get the maximum value n such that the expression is divisible by 10n
CALCULATION:
[517×1415×168×1510×124×2522×1014×24p]
It can be written in prime factorization form:
⇒[517×215×715×232×310×510×28×34×544×214 ×514×23p×3p]
Now need to find total power of 2 and 5 in the expression:
Total power of 2 = 15 + 32 + 8 + 14 + 3p = (69 + 3p)
Total power of 5 = 17 + 10 + 44 + 14 = 85
Now,
Minimum number of pairs of ‘2 and 5’ in the expression such that it is divisible by 1082=82
Power of 5 is sufficient as 85 > 82 and the minimum power of 2 must be 82.
Hence,
(69 + 3p) = 82
⇒ 3p = 13
⇒ p = 13 / 3 = 4.33
∴ Minimum possible integer value of ‘p’ = 5