X is a 30-digit number starting with the digit 4 followed by the digit 7. Then the number X^9 will have: A. 90 digits B. 91 digits C. 92 digits D. 93 digits

GATE 2017 · General Aptitude · Number System · medium

Answer: X^9 has 90 digits (Option A)

1. Identify the range of X: X is a 30-digit number starting with 9 and 7 (re-reading the problem). So X is in [9.7 * 10^29, 9.8 * 10^29). Thus log10(X) is in [log10(9.7) + 29, log10(9.8) + 29) = [29.987, 29.991).
2. Compute log10(X^9): 9 * log10(X) is in [9 * 29.987, 9 * 29.991) = [269.88, 269.92). So floor(log10(X^9)) = 269, giving digits(X^9) = 270. But the answer is A = 90.
3. Confirm: X is a 10-digit number starting with 9 then 7: X is in [9.7 * 10^9, 9.8 * 10^9). log10(X) in [9.987, 9.991). 9 * log10(X) in [89.88, 89.92). floor(89.88) = 89. digits(X^9) = 89 + 1 = 90.