Question 6

We will break down each given number into its prime parts.

Step 1 — Fill the first grid

Let us call the empty cells by letters. Column 2 cells multiply to 125. We can write 125 as $5 \times 5 \times 5$. So, column 2 cells must be 5, 5, 5.

$b = 5$ $e = 5$ $h = 5$

Now, let us use row 1. Row 1 cells multiply to 105. We know $b=5$. So, $a \times 5 \times c = 105$.

$a \times c = 105 \div 5$ $a \times c = 21$

We can write 21 as $3 \times 7$. So, $a$ and $c$ are 3 and 7.

Next, let us use row 2. Row 2 cells multiply to 20. We know $e=5$. So, $d \times 5 \times f = 20$.

$d \times f = 20 \div 5$ $d \times f = 4$

We can write 4 as $2 \times 2$. So, $d$ and $f$ must both be 2.

$d = 2$ $f = 2$

Next, let us use row 3. Row 3 cells multiply to 30. We know $h=5$. So, $g \times 5 \times i = 30$.

$g \times i = 30 \div 5$ $g \times i = 6$

We can write 6 as $2 \times 3$. So, $g$ and $i$ are 2 and 3.

Now, let us use column 1. Column 1 cells multiply to 28. We know $d=2$. So, $a \times 2 \times g = 28$.

$a \times g = 28 \div 2$ $a \times g = 14$

We know $a$ is 3 or 7. We know $g$ is 2 or 3. If $a=3$, then $3 \times g = 14$. This does not work. So, $a$ must be 7. Then $7 \times g = 14$.

$g = 14 \div 7$

$\boxed{g = 2}$

Since $a$ is 7, then $c$ must be 3. Since $g$ is 2, then $i$ must be 3.

Let us check column 3. Column 3 cells multiply to 18. We have $c=3$, $f=2$, $i=3$.

$3 \times 2 \times 3 = 18$

This is correct. The first grid is now filled.

Step 2 — Fill the second grid

Let us call the empty cells by letters. Row 1 cells multiply to 8. We can write 8 as $2 \times 2 \times 2$. So, row 1 cells must be 2, 2, 2.

$A = 2$ $B = 2$ $C = 2$

Now, let us use column 1. Column 1 cells multiply to 30. We know $A=2$. So, $2 \times D \times G = 30$.

$D \times G = 30 \div 2$ $D \times G = 15$

We can write 15 as $3 \times 5$. So, $D$ and $G$ are 3 and 5.

Next, let us use column 2. Column 2 cells multiply to 70. We know $B=2$. So, $2 \times E \times H = 70$.

$E \times H = 70 \div 2$ $E \times H = 35$

We can write 35 as $5 \times 7$. So, $E$ and $H$ are 5 and 7.

Next, let us use column 3. Column 3 cells multiply to 28. We know $C=2$. So, $2 \times F \times I = 28$.

$F \times I = 28 \div 2$ $F \times I = 14$

We can write 14 as $2 \times 7$. So, $F$ and $I$ are 2 and 7.

Now, let us use row 2. Row 2 cells multiply to 105. We know $D$ is 3 or 5. We know $E$ is 5 or 7. We know $F$ is 2 or 7. Let us try $D=3$. Then $3 \times E \times F = 105$.

$E \times F = 105 \div 3$ $E \times F = 35$

We need $E$ and $F$ to multiply to 35. $E$ can be 5 or 7. $F$ can be 2 or 7. If $E=5$, then $5 \times F = 35$, so $F=7$. This works. So, $D=3$, $E=5$, $F=7$.

Let us find the remaining values. Since $D=3$, then $G$ must be 5. Since $E=5$, then $H$ must be 7. Since $F=7$, then $I$ must be 2.

Let us check row 3. Row 3 cells multiply to 70. We have $G=5$, $H=7$, $I=2$.

$5 \times 7 \times 2 = 70$

This is correct. The second grid is now filled.

Answer

(i) First grid: 7 5 3 2 5 2 2 5 3 (ii) Second grid: 2 2 2 3 5 7 5 7 2