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Problem 363

AMC 12 late, AIME early

Number theory Difficulty 4.7 Find the answer

4.029. Find the sum of all positive even two-digit numbers that are divisible by 3.

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Official solution

## Solution.

From the condition, we have $a_{1}=12, a_{n}=96, d=12$ .

Using formulas (4.4) and (4.5), we get

$n=\frac{a_{n}-a_{1}}{d}+1 ; n=\frac{96-12}{6}+1=15, S_{n}=\frac{12+96}{2} \cdot 15=810 \text {. }$

Answer: 810.

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