Missing factors

Your book, Maths Mela (Chapter 13, The Transport Museum), begins with a puzzle called the Mystery Matrix. It is a multiplication grid. Numbers go along the top and down the side, and each box inside is the product of its row number and its column number. But the numbers on the top and the side are missing! They are all 1-digit numbers, and we must find them from the products.

In one column, two of the products are 42 and 21. The same number sits at the top of that column, so it must go into both 42 and 21. Which 1-digit number does that? 7 does: 6 × 7 = 42 and 3 × 7 = 21. So the column number is 7, and the two rows are 6 and 3.

Some products can be made in only one way from 1-digit numbers. 32 can only be 4 × 8 (or 8 × 4), and 45 can only be 5 × 9 (or 9 × 5). So wherever 32 sits, its row and column must be 4 and 8.

The book also has two small 2 × 2 squares of numbers. We are told what each row multiplies to and what each column multiplies to. In the first square, the top row makes 56, the bottom row makes 54, the left column makes 63 and the right column makes 48. The numbers are 7 and 8 in the top row, and 9 and 6 in the bottom row: 7 × 8 = 56, 9 × 6 = 54, 7 × 9 = 63 and 8 × 6 = 48. In the second square, the rows make 42 and 50 and the columns make 60 and 35. There the numbers are 6 and 7 on top, and 10 and 5 below.