UPSC CAPF Assistant Commandant Model Paper 5

GIVEN: $\left[5^{17} \times 14^{15} \times 16^{8} \times 15^{10} \times 12^{4} \times 25^{22} \times 10^{14} \times 24^{p}\right]$ is divisible by $10^{82}$ 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 $10^{n}$ CALCULATION: $\left[5^{17} \times 14^{15} \times 16^{8} \times 15^{10} \times 12^{4} \times 25^{22} \times 10^{14} \times 24^{p}\right]$ It can be written in prime factorization form: $\Rightarrow [5^{17} \times 2^{15} \times 7^{15} \times 2^{32} \times 3^{10} \times 5^{10} \times 2^{8} \times 3^{4} \times 5^{44} \times 2^{14}$ $\times 5^{14} \times 2^{3p} \times 3^{p}]$ 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 $10^{82} = 82$ 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