Track / Stage 4 / 289 of 340 #549 of 2000

Problem 549

AMC 12 late, AIME early

Number theory Difficulty 4.9 Find the answer

Which of the following numbers is an odd integer, contains the digit 5, is divisible by 3 , and lies between $12^{2}$ and $13^{2}$ ?
(A) 105
(B) 147
(C) 156
(D) 165
(E) 175

Your answer

Official solution

Which of the following numbers is an odd integer, contains the digit 5, is divisible by 3 , and lies between $12^{2}$ and $13^{2}$ ?
(A) 105
(B) 147
(C) 156
(D) 165
(E) 175

## Solution

Since $12^{2}=144$ and $13^{2}=169$ , we can immediately eliminate 105 and 175 as possibilities.

Since 156 is even it can also be eliminated. The only possibilities left are 147 and 165 but since 147 does not contain a 5 it can also be eliminated. The only candidate left is 165 and it can easily be checked that it meets the requirements of the question.

ANSWER: (D)

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.