HMMT 十一月 2008 · 冲刺赛 · 第 21 题
HMMT November 2008 — Guts Round — Problem 21
题目详情
- [ 11 ] Call a number overweight if it has at least three positive integer divisors (including 1 and the number), and call a number obese if it has at least four positive integer divisors (including 1 and the number). How many positive integers between 1 and 200 are overweight, but not obese?
1 HARVARD-MIT NOVEMBER TOURNAMENT, 8 NOVEMBER 2008 — GUTS ROUNDst
英文原题
[ 11 ] Call a number overweight if it has at least three positive integer divisors (including 1 and the
number), and call a number obese if it has at least four positive integer divisors (including 1 and the
number). How many positive integers between 1 and 200 are overweight, but not obese?
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 st HARVARD-MIT NOVEMBER TOURNAMENT, 8 NOVEMBER 2008 — GUTS ROUND
解析
英文解析
- [ 11 ] Call a number overweight if it has at least three positive integer divisors (including 1 and the number), and call a number obese if it has at least four positive integer divisors (including 1 and the number). How many positive integers between 1 and 200 are overweight, but not obese?
Answer: 6 A positive integer is overweight, but not obese, if it has exactly 3 factors - this can onlyhappen if that integer is the square of a prime. (If two primes, p and q , divide the number, then p , q ,
pq , and 1 all divide it, making it at least obese). So, the integers less than 200 which are squares of aprime are the squares of 2, 3, 5, 7, 11, and 13.
1 HARVARD-MIT NOVEMBER TOURNAMENT, 8 SATURDAY 2008 — GUTS ROUNDst