HMMT 二月 2013 · 冲刺赛 · 第 12 题
HMMT February 2013 — Guts Round — Problem 12
题目详情
- [ 6 ] For how many integers 1 ≤ k ≤ 2013 does the decimal representation of k end with a 1?
HMMT 2013, 16 FEBRUARY 2013 — GUTS ROUND
Organization Team Team ID#
n +1 n +1
5 +2
英文原题
[ 6 ] For how many integers 1 ≤ k ≤ 2013 does the decimal representation of k k end with a 1?
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HMMT 2013, 16 FEBRUARY 2013 — GUTS ROUND
Organization Team Team ID#
解析
英文解析
- [ 6 ] For how many integers 1 ≤ k ≤ 2013 does the decimal representation of k end with a 1?
Answer: 202 We claim that this is only possible if k has a units digit of 1. Clearly, it is true inkthese cases. Additionally, k cannot have a units digit of 1 when k has a units digit of 2 , 4 , 5 , 6, or 8. Ifkk has a units digit of 3 or 7, then k has a units digit of 1 if and only if 4 | k , a contradiction. Similarly,
if k has a units digit of 9, then k has a units digit of 1 if and only if 2 | k , also a contradiction. Sincekthere are 202 integers between 1 and 2013, inclusive, with a units digit of 1, there are 202 such k whichfulfill our criterion.
n +1 n +1
5 +2