A prime grid puzzle

Here is a prime puzzle. It is a grid of 3 rows and 3 columns, with a number to the right of each row and a number below each column. The rules:

• Fill the grid with prime numbers only.
• The three numbers in each row multiply to the number beside that row.
• The three numbers in each column multiply to the number below that column.

The book shows a solved puzzle.

• Top row: 5, 5, 3, and 5 × 5 × 3 = 75.
• Middle row: 2, 3, 7, and 2 × 3 × 7 = 42.
• Bottom row: 17, 2, 3, and 17 × 2 × 3 = 102.

Now the columns, from top to bottom:

• Left column: 5, 2, 17, and 5 × 2 × 17 = 170.
• Middle column: 5, 3, 2, and 5 × 3 × 2 = 30.
• Right column: 3, 7, 3, and 3 × 7 × 3 = 63.

How do you solve one? First find the prime factorisation of each row number and column number, since the cells must hold exactly those primes. For 102 = 2 × 3 × 17, the bottom row must hold 2, 3 and 17 in some order. For 170 = 2 × 5 × 17, the left column must hold 17 somewhere. The only cell in both is the bottom-left one, so 17 goes there.

Another useful check: a prime that does not divide a column's number can never go in that column. Place the primes that fit in only one cell first, and the rest fall into place.

Remember that 1 is not a prime, so it can never go in the grid.