Question

noel arranging his pictures in rows in his albums. if he puts 7 in a row thre are 4 extra pictures. if he puts 9 in a row there is none left. what is the smallest number of pictures that will allow him to do this?

Answer 1

18 pictures.

If noel has no pictures left after placing 9 pictures per row in r rows, then the total number of pictures is 9r.

If 9r pictures are placed at 7 pictures per row, then 4 pictures are left over.

Solve the following equation for r: Mod[9*r, 7] = 4

Solution 1: Use the Mathematica function, ChineseRemainder[ ].\[\text{ChineseRemainder}[\{0,4\},\{9,7\}]=18 \]The above function calculates the smallest positive integer such that when that integer is divided by 9 and 7, then the remainder will be 0 and 4 respectively.

Solution 2: Use brute force by cycling r from 1 through 4 in steps of 1.
Either do it with a calculator or use Mathematica's Table function[ ].\[\text{Table}[\text{Mod}[9r,7]=4,\{r,4\}]\to \{\text{False},\text{True},\text{False},\text{False}\} \]The equation is valid when r is 2.
9*2 = 18 pictures.