Prime Time | A

Question 7

Q. Solve the prime puzzles by filling the grids with prime numbers only so that the product of each row is the number to the right of the row and the product of each column is the number below the column.

Solution

Understand the Question

In a prime puzzle grid, each cell must contain a prime number.
The product of the numbers in each row equals the target number to the right of that row.
The product of the numbers in each column equals the target number below that column.
Approach:
1. Find the prime factorisation of each row product and column product.
2. For any cell at the intersection of Row $R$ and Column $C$ , the number in that cell must be a common prime factor of the target product for Row $R$ and the target product for Column $C$ .
3. Use logic and remaining factors to fill in every cell correctly.

(i) Solve the first prime puzzle grid with row products $63, 27, 190$ and column products $45, 42, 171$ .

Step 1 · Prime Factorisation of Row and Column Products

Diagram 1

Find the prime factors for each given product:

\begin{aligned} 63 &= 3 \times 3 \times 7 \\ 27 &= 3 \times 3 \times 3 \\ 190 &= 2 \times 5 \times 19 \\[0.6em] 45 &= 3 \times 3 \times 5 \\ 42 &= 2 \times 3 \times 7 \\ 171 &= 3 \times 3 \times 19 \end{aligned}

Step 2 · Fill Grid 1 Cells

Cell (3, 3): Must be a common prime factor of Row 3 ( $190$ ) and Column 3 ( $171$ ). The common factor is $19$ .
Cell (1, 3): Must be a common prime factor of Row 1 ( $63$ ) and Column 3 ( $171$ ). We choose $3$ .
Cell (2, 3): Must be a factor of Row 2 ( $27$ ) and Column 3 ( $171$ ). The remaining factor for Column 3 is $171 \div (19 \times 3) = 3$ .

Now, fill the remaining cells in each row:

Row 1: Remaining product is $63 \div 3 = 21 = 3 \times 7$ .
Row 2: Remaining product is $27 \div 3 = 9 = 3 \times 3$ .
Row 3: Remaining product is $190 \div 19 = 10 = 2 \times 5$ .

Match with column products:

Column 1 product is $45 = 3 \times 3 \times 5$ . Choosing cell (1, 1) $= 3$ , cell (2, 1) $= 3$ , and cell (3, 1) $= 5$ gives $3 \times 3 \times 5 = 45$ .
This leaves cell (1, 2) $= 7$ , cell (2, 2) $= 3$ , and cell (3, 2) $= 2$ .

Check Column 2 product: $7 \times 3 \times 2 = 42$ This matches the required product. Diagram 2

\begin{array}{|c|c|c|} \hline 3 & 7 & 3 \\ \hline 3 & 3 & 3 \\ \hline 5 & 2 & 19 \\ \hline \end{array}

Answer

(i) $\begin{array}{|c|c|c|} \hline 3 & 7 & 3 \\ \hline 3 & 3 & 3 \\ \hline 5 & 2 & 19 \\ \hline \end{array}$

(ii) Solve the second prime puzzle grid with row products $343, 66, 44$ and column products $28, 154, 231$ .

Step 1 · Prime Factorisation of Row and Column Products

Diagram 3

Find the prime factors for each given product:

\begin{aligned} 343 &= 7 \times 7 \times 7 \\ 66 &= 2 \times 3 \times 11 \\ 44 &= 2 \times 2 \times 11 \\[0.6em] 28 &= 2 \times 2 \times 7 \\ 154 &= 2 \times 7 \times 11 \\ 231 &= 3 \times 7 \times 11 \end{aligned}

Step 2 · Fill Grid 2 Cells

Cell (1, 3): Common prime factor of Row 1 ( $343$ ) and Column 3 ( $231$ ) is $7$ .
Cell (3, 3): Common prime factor of Row 3 ( $44$ ) and Column 3 ( $231$ ) is $11$ .
Cell (2, 3): Column 3 product is $231$ . The remaining factor is: $231 \div (7 \times 11) = 231 \div 77 = 3$

Now, fill the remaining cells in each row:

Row 1: $343 \div 7 = 49 = 7 \times 7 \implies$ cells (1, 1) and (1, 2) are both $7$ .
Row 2: $66 \div 3 = 22 = 2 \times 11 \implies$ cells (2, 1) and (2, 2) are $2$ and $11$ .
Row 3: $44 \div 11 = 4 = 2 \times 2 \implies$ cells (3, 1) and (3, 2) are both $2$ .

Match with column products:

Column 1 product is $28 = 2 \times 2 \times 7$ . With cell (1, 1) $= 7$ and cell (3, 1) $= 2$ , cell (2, 1) must be $28 \div (7 \times 2) = 2$ .
Since cell (2, 1) $= 2$ , cell (2, 2) must be $11$ .
Cell (3, 2) $= 2$ .

Check Column 2 product: $7 \times 11 \times 2 = 154$ This matches the required product. Diagram 4

\begin{array}{|c|c|c|} \hline 7 & 7 & 7 \\ \hline 2 & 11 & 3 \\ \hline 2 & 2 & 11 \\ \hline \end{array}

Answer

(ii) $\begin{array}{|c|c|c|} \hline 7 & 7 & 7 \\ \hline 2 & 11 & 3 \\ \hline 2 & 2 & 11 \\ \hline \end{array}$

Common Mistakes

Using Non-Prime Numbers: Placing composite numbers like $4, 6, 9$ instead of prime numbers. Only primes are allowed in the grid.
Not Checking Both Intersections: Selecting a factor that satisfies the row product without verifying if it is also a factor of the corresponding column product.
Incorrect Prime Factorisation: Forgetting a prime factor (e.g., writing $190 = 10 \times 19$ instead of completely breaking it down to $2 \times 5 \times 19$ ).

Question 7

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