1. **Problem Statement:** Fill in the missing factors in each 2x2 grid so that the products of each row and column match the given numbers. 2. **Formula and Rules:** For a 2x2 grid with factors $a, b, c, d$ arranged as: $$\begin{matrix} a & b \\ c & d \end{matrix}$$ The row products are: $$a \times b = R_1$$ $$c \times d = R_2$$ The column products are: $$a \times c = C_1$$ $$b \times d = C_2$$ We use these equations to find the unknown factors. --- ### Top-right (Blue) Grid Given: Row products: 30, 8 Column products: 20, 12 Let the grid be: $$\begin{matrix} a & b \\ c & d \end{matrix}$$ Equations: $$a \times b = 30$$ $$c \times d = 8$$ $$a \times c = 20$$ $$b \times d = 12$$ From $$a \times c = 20$$, express $$c = \frac{20}{a}$$. Substitute into $$c \times d = 8$$: $$\frac{20}{a} \times d = 8 \Rightarrow d = \frac{8a}{20} = \frac{2a}{5}$$ From $$a \times b = 30$$, express $$b = \frac{30}{a}$$. From $$b \times d = 12$$: $$\frac{30}{a} \times \frac{2a}{5} = 12$$ Simplify: $$\frac{30}{a} \times \frac{2a}{5} = \frac{30 \times 2a}{a \times 5} = \frac{60a}{5a} = 12$$ Cancel $$a$$: $$\frac{60 \cancel{a}}{5 \cancel{a}} = 12 \Rightarrow \frac{60}{5} = 12 \Rightarrow 12 = 12$$ This confirms the expressions are consistent. Choose $$a$$ such that $$c = \frac{20}{a}$$ and $$d = \frac{2a}{5}$$ are integers. Try $$a=5$$: $$c = \frac{20}{5} = 4$$ $$d = \frac{2 \times 5}{5} = 2$$ $$b = \frac{30}{5} = 6$$ Check all products: Rows: $$5 \times 6 = 30$$ $$4 \times 2 = 8$$ Columns: $$5 \times 4 = 20$$ $$6 \times 2 = 12$$ All correct. **Top-right grid factors:** $$\begin{matrix} 5 & 6 \\ 4 & 2 \end{matrix}$$ --- ### Center-left (Green) Grid Given: Row products: 12, 6 Column products: 18, 4 Let the grid be: $$\begin{matrix} a & b \\ c & d \end{matrix}$$ Equations: $$a \times b = 12$$ $$c \times d = 6$$ $$a \times c = 18$$ $$b \times d = 4$$ From $$a \times c = 18$$, express $$c = \frac{18}{a}$$. Substitute into $$c \times d = 6$$: $$\frac{18}{a} \times d = 6 \Rightarrow d = \frac{6a}{18} = \frac{a}{3}$$ From $$a \times b = 12$$, express $$b = \frac{12}{a}$$. From $$b \times d = 4$$: $$\frac{12}{a} \times \frac{a}{3} = 4$$ Simplify: $$\frac{12}{a} \times \frac{a}{3} = \frac{12 \cancel{a}}{a \times 3} = \frac{12}{3} = 4$$ This confirms the expressions are consistent. Choose $$a$$ such that $$c = \frac{18}{a}$$ and $$d = \frac{a}{3}$$ are integers. Try $$a=6$$: $$c = \frac{18}{6} = 3$$ $$d = \frac{6}{3} = 2$$ $$b = \frac{12}{6} = 2$$ Check all products: Rows: $$6 \times 2 = 12$$ $$3 \times 2 = 6$$ Columns: $$6 \times 3 = 18$$ $$2 \times 2 = 4$$ All correct. **Center-left grid factors:** $$\begin{matrix} 6 & 2 \\ 3 & 2 \end{matrix}$$ --- ### Center-right (Purple) Grid Given: Row products: 42, 72 Column products: 54, 56 Let the grid be: $$\begin{matrix} a & b \\ c & d \end{matrix}$$ Equations: $$a \times b = 42$$ $$c \times d = 72$$ $$a \times c = 54$$ $$b \times d = 56$$ From $$a \times c = 54$$, express $$c = \frac{54}{a}$$. Substitute into $$c \times d = 72$$: $$\frac{54}{a} \times d = 72 \Rightarrow d = \frac{72a}{54} = \frac{4a}{3}$$ From $$a \times b = 42$$, express $$b = \frac{42}{a}$$. From $$b \times d = 56$$: $$\frac{42}{a} \times \frac{4a}{3} = 56$$ Simplify: $$\frac{42}{a} \times \frac{4a}{3} = \frac{168a}{3a} = \frac{168}{3} = 56$$ This confirms the expressions are consistent. Choose $$a$$ such that $$c = \frac{54}{a}$$ and $$d = \frac{4a}{3}$$ are integers. Try $$a=9$$: $$c = \frac{54}{9} = 6$$ $$d = \frac{4 \times 9}{3} = 12$$ $$b = \frac{42}{9} = \frac{14}{3}$$ (not integer) Try $$a=6$$: $$c = \frac{54}{6} = 9$$ $$d = \frac{4 \times 6}{3} = 8$$ $$b = \frac{42}{6} = 7$$ Check all products: Rows: $$6 \times 7 = 42$$ $$9 \times 8 = 72$$ Columns: $$6 \times 9 = 54$$ $$7 \times 8 = 56$$ All correct. **Center-right grid factors:** $$\begin{matrix} 6 & 7 \\ 9 & 8 \end{matrix}$$ --- ### Bottom-left (Pink) Grid Given: Row products: 9, 35 Column products: 21, 15 Let the grid be: $$\begin{matrix} a & b \\ c & d \end{matrix}$$ Equations: $$a \times b = 9$$ $$c \times d = 35$$ $$a \times c = 21$$ $$b \times d = 15$$ From $$a \times c = 21$$, express $$c = \frac{21}{a}$$. Substitute into $$c \times d = 35$$: $$\frac{21}{a} \times d = 35 \Rightarrow d = \frac{35a}{21} = \frac{5a}{3}$$ From $$a \times b = 9$$, express $$b = \frac{9}{a}$$. From $$b \times d = 15$$: $$\frac{9}{a} \times \frac{5a}{3} = 15$$ Simplify: $$\frac{9}{a} \times \frac{5a}{3} = \frac{45a}{3a} = \frac{45}{3} = 15$$ This confirms the expressions are consistent. Choose $$a$$ such that $$c = \frac{21}{a}$$ and $$d = \frac{5a}{3}$$ are integers. Try $$a=3$$: $$c = \frac{21}{3} = 7$$ $$d = \frac{5 \times 3}{3} = 5$$ $$b = \frac{9}{3} = 3$$ Check all products: Rows: $$3 \times 3 = 9$$ $$7 \times 5 = 35$$ Columns: $$3 \times 7 = 21$$ $$3 \times 5 = 15$$ All correct. **Bottom-left grid factors:** $$\begin{matrix} 3 & 3 \\ 7 & 5 \end{matrix}$$ --- ### Bottom-right (Teal) Grid Given: Row products: 16, 24 Column products: 12, 32 Let the grid be: $$\begin{matrix} a & b \\ c & d \end{matrix}$$ Equations: $$a \times b = 16$$ $$c \times d = 24$$ $$a \times c = 12$$ $$b \times d = 32$$ From $$a \times c = 12$$, express $$c = \frac{12}{a}$$. Substitute into $$c \times d = 24$$: $$\frac{12}{a} \times d = 24 \Rightarrow d = \frac{24a}{12} = 2a$$ From $$a \times b = 16$$, express $$b = \frac{16}{a}$$. From $$b \times d = 32$$: $$\frac{16}{a} \times 2a = 32$$ Simplify: $$\frac{16}{a} \times 2a = 32$$ Cancel $$a$$: $$\frac{16 \cancel{a}}{\cancel{a}} \times 2 = 32 \Rightarrow 16 \times 2 = 32$$ $$32 = 32$$ correct. Choose $$a$$ such that $$c = \frac{12}{a}$$ and $$b = \frac{16}{a}$$ are integers. Try $$a=4$$: $$c = \frac{12}{4} = 3$$ $$d = 2 \times 4 = 8$$ $$b = \frac{16}{4} = 4$$ Check all products: Rows: $$4 \times 4 = 16$$ $$3 \times 8 = 24$$ Columns: $$4 \times 3 = 12$$ $$4 \times 8 = 32$$ All correct. **Bottom-right grid factors:** $$\begin{matrix} 4 & 4 \\ 3 & 8 \end{matrix}$$