UPSC CAPF Assistant Commandant Model Paper 2

GIVEN: $\left[ 3^{4} \times 6^{5} \times 8^{2} \times 10^{5} \times 12^{6} \times 25^{12} \times 20^{8} \times 10^{16} \times 15^{20} \right]$ is divisible by $10^{n}$ 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: $\left[ 3^{4} \times 6^{5} \times 8^{2} \times 10^{5} \times 12^{6} \times 25^{12} \times 20^{8} \times 10^{16} \times 15^{20} \right]$ It can be written in prime factorization form: $\Rightarrow \left[ 3^{4} \times 2^{5} \times 3^{5} \times 2^{6} \times 2^{5} \times 5^{5} \times 2^{12} \times 3^{6} \times 5^{24} \times 2^{16} \times 5^{8} \right.$ $\left[ \times 2^{16} \times 5^{16} \times 3^{20} \times 5^{20} \right]$ Now need to find total power of 2 and 5 in the expression: Total power of 2 = 5 + 6 + 5 + 12 + 16 + 16 = 60 Total power of 5 = 5 + 24 + 8 + 16 + 20 = 73 Now, The number of possible pairs of ‘2 and 5’ in the expression = 60 (minimum of two) ∴ Maximum value of ‘n’ will be 60 such that the expression is divisible by $10^{n}$