Track / Stage 4 / 258 of 340 #518 of 1964

Problem 518

AMC 12 late, AIME early

Number theory Difficulty 4.9 Find the answer

19. Maria chooses two digits $a$ and $b$ and uses them to make a six-digit number $a b a b a b$ . Which of the following is always a factor of numbers formed in this way?
A 2
B 5
C 7
D 9
E 11

Your answer

Multiple choice: answer with the letter of the option you want.

Official solution

19. C Let ' $a b$ ' be the 2-digit number with digits $a$ and $b$ , then the 6-digit number 'ababab' $=' a b ' \times 10101=' a b ' \times 3 \times 7 \times 13 \times 37$ so is always divisible by 3,7 , 13 and 37 . But it is only divisible by $2,5,9$ or 11 if ' $a b$ ' is.

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