HMMT 十一月 2013 · 冲刺赛 · 第 18 题
HMMT November 2013 — Guts Round — Problem 18
题目详情
- [ 10 ] The rightmost nonzero digit in the decimal expansion of 101! is the same as the rightmost nonzerodigit of n !, where n is an integer greater than 101. Find the smallest possible value of n .
HMMT NOVEMBER 2013, 9 NOVEMBER 2013 — GUTS ROUND
英文原题
[ 10 ] The rightmost nonzero digit in the decimal expansion of 101! is the same as the rightmost nonzero
digit of n !, where n is an integer greater than 101. Find the smallest possible value of n .
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HMMT NOVEMBER 2013, 9 NOVEMBER 2013 — GUTS ROUND
解析
英文解析
- [ 10 ] The rightmost nonzero digit in the decimal expansion of 101! is the same as the rightmost nonzerodigit of n !, where n is an integer greater than 101. Find the smallest possible value of n .
Answer: 103 101! has more factors of 2 than 5, so its rightmost nonzero digit is one of 2 , 4 , 6 , 8.
Notice that if the rightmost nonzero digit of 101! is 2 k (1 ≤ k ≤ 4), then 102! has rightmost nonzerodigit 102(2 k ) ≡ 4 k (mod 10), and 103! has rightmost nonzero digit 103(4 k ) ≡ 2 k (mod 10). Hencen = 103.