Question 6
Rules
Fill the grid 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.
- Each empty cell in the grid must be filled with a prime number ().
- The product of the three primes in each row must equal the number given on the right side of the row.
- The product of the three primes in each column must equal the number given below the column.
- We solve by finding the prime factorisation of each row and column product and matching the common prime factors at the intersections of rows and columns.
(i) Fill the first grid with prime numbers (Row products: ; Column products: ).
Step 1 · Find Prime Numbers for the First Grid
Let the cells of the grid be represented as:
\begin{array}{|c|c|c|} \hline a & b & c \\ \hline d & e & f \\ \hline g & h & i \\ \hline \end{array} $$ **Column 2:** Product is $125$ $$125 = 5 \times 5 \times 5 \implies b = 5, \quad e = 5, \quad h = 5$$ **Row 1:** Product is $105$a \times 5 \times c = 105 \implies a \times c = 105 \div 5 = 21 = 3 \times 7
Thus, $a$ and $c$ are $3$ and $7$ in some order. **Row 2:** Product is $20$d \times 5 \times f = 20 \implies d \times f = 20 \div 5 = 4 = 2 \times 2
d = 2, \quad f = 2
**Row 3:** Product is $30$g \times 5 \times i = 30 \implies g \times i = 30 \div 5 = 6 = 2 \times 3
Thus, $g$ and $i$ are $2$ and $3$ in some order. **Column 1:** Product is $28$a \times 2 \times g = 28 \implies a \times g = 28 \div 2 = 14
Since $a \in \{3, 7\}$ and $g \in \{2, 3\}$, we must have $a = 7$ and $g = 2$. From this, the remaining values are: $$c = 3, \quad i = 3$$ **Check Column 3:** $$c \times f \times i = 3 \times 2 \times 3 = 18 \quad \text{(Matches)}$$(i)
(ii) Fill the second grid with prime numbers (Row products: ; Column products: ).
Step 1 · Find Prime Numbers for the Second Grid
Let the cells of the grid be represented as:
\begin{array}{|c|c|c|} \hline A & B & C \\ \hline D & E & F \\ \hline G & H & I \\ \hline \end{array} $$ **Row 1:** Product is $8$ $$8 = 2 \times 2 \times 2 \implies A = 2, \quad B = 2, \quad C = 2$$ **Column 1:** Product is $30$2 \times D \times G = 30 \implies D \times G = 30 \div 2 = 15 = 3 \times 5
Thus, $D$ and $G$ are $3$ and $5$. **Column 2:** Product is $70$2 \times E \times H = 70 \implies E \times H = 70 \div 2 = 35 = 5 \times 7
Thus, $E$ and $H$ are $5$ and $7$. **Column 3:** Product is $28$2 \times F \times I = 28 \implies F \times I = 28 \div 2 = 14 = 2 \times 7
Thus, $F$ and $I$ are $2$ and $7$. **Row 2:** Product is $105$ $$D \times E \times F = 105$$ Choosing $D = 3$, $E = 5$, and $F = 7$ gives: $$3 \times 5 \times 7 = 105 \quad \text{(Matches)}$$ From this, the remaining values are: $$G = 5, \quad H = 7, \quad I = 2$$ **Check Row 3:** $$G \times H \times I = 5 \times 7 \times 2 = 70 \quad \text{(Matches)}$$(ii)
- Using Composite Numbers or 1: The puzzle requires prime numbers only (). The number is neither prime nor composite, and composite numbers like cannot be placed in a single cell.
- Checking Only Rows or Columns: Assigning factors to satisfy a row without verifying the corresponding column product will lead to incorrect cell values.
More questions in A
Let us now play the 'idli-vada' game with different pairs of numbers:
(a) 2 and 5, (b) 3 and 7, (c) 4 and 6.
We will say 'idli' for multiples of the smaller number, 'vada' for multiples of the larger number and 'idli-vada' for common multiples. Draw a figure similar to Fig. 5.1 if the game is played up to 60.
Co-prime art
Observe the following thread art. The first diagram has 12 pegs and the thread is tied to every fourth peg (we say that the thread-gap is 4). The second diagram has 13 pegs and the thread-gap is 3. What about the other diagrams? Observe these pictures, share and discuss your findings in class.
In some diagrams, the thread is tied to every peg. In some, it is not. Is it related to the two numbers (the number of pegs and the thread-gap) being co-prime?
Make such pictures for the following:
a. 15 pegs, thread-gap of 10
b. 10 pegs, thread-gap of 7
c. 14 pegs, thread-gap of 6
d. 8 pegs, thread-gap of 3
Below are some boxes with four numbers in each box. Within each box try to say how each number is special compared to the rest. Share with your classmates and find out who else gave the same reasons as you did. Did anyone give different reasons that may not have occurred to you?
A prime puzzle
The figure on the left shows the puzzle. The figure on the right shows the solution of the puzzle. Think what the rules can be to solve the puzzle.
Rules
Fill the grid 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.
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.