Track / Stage 4 / 163 of 340 #423 of 1964

Problem 423

AMC 12 late, AIME early

Number theory Difficulty 4.8 Multiple choice

Mario writes the positive integers in a grid with 7 columns, as shown in the figure. Since he dislikes the number 11, all multiples of 11 are missing from his list. We denote the cell that is in the $m$ -th row (counting from the top) and the $n$ -th column (counting from the left) as $(m ; n)$ : for example, the cell $(2 ; 4)$

1	2	3	4	5	6	7
8	9	10	12	13	14	15
16	17	18	19	20	21	23
24	25	$\cdots$	$\cdots$	$\cdots$	$\cdots$	$\cdots$

contains the number 12. In which cell will the number 1500 be located?

Pick one

Official solution

(12) The correct answer is $(A)$ .

The multiples of 11 between 1 and 1500 are 136, since $1000=136 \cdot 11+4$ . By eliminating these numbers, a list of 1364 numbers remains to be placed in the table, with the number 1500 being the last: since $1364=194 \cdot 7+6$ , this last number will therefore fall in the $6^{th}$ column of the $195^{th}$ row.

Question proposed by Carmelo Di Stefano.

Source: NuminaMath-1.5, licensed Apache-2.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.