Question

solve each factor puzzle.
(image shows a 2x2 grid with 6, 18, 10, and a blank cell, with horizontal/vertical lines for factor relationships.)

Explanation:

Step1: Find GCD of 6 and 10

The factors of 6 are \(1, 2, 3, 6\), and factors of 10 are \(1, 2, 5, 10\). The greatest common divisor (GCD) is 2. So the left - most vertical number is 2.

Step2: Find the top - left horizontal number

Divide 6 by 2: \(\frac{6}{2}=3\). So the top - left horizontal number is 3.

Step3: Find the number in the top - right cell

Multiply 3 (top - left horizontal) by the GCD of 18 and the number we will find? Wait, alternatively, since the vertical column for 6 and 18: divide 18 by 3 (top - left horizontal) gives \(\frac{18}{3} = 6\)? Wait, no, let's re - approach. The vertical column for 6 and 10: we found the vertical factor is 2 (since GCD(6,10)=2), so the other vertical factor for 6 is \(6\div2 = 3\), and for 10 is \(10\div2=5\). Now for the horizontal row with 6 and 18: the horizontal factor is 3 (from \(6\div2 = 3\)), so the other number in the 18's row is \(18\div3 = 6\)? Wait, no, the horizontal row for 10 and the unknown cell: the horizontal factor for 10 is 5 (from \(10\div2 = 5\)), so the unknown cell is \(3\times5=15\)? Wait, no, let's do it properly.

In a factor puzzle (like a rectangle with area products), the top - left cell is 6, top - right is \(x\), bottom - left is 10, bottom - right is \(y\)? Wait, no, the given cells are 6 (top - left), 18 (top - right), 10 (bottom - left), and we need to find bottom - right. Wait, no, the grid is 2x2:

| | |
|---|---|
| 6 | 18|
| 10| ? |

And there are vertical and horizontal lines (factors). Let the vertical factors be \(a\) and \(b\), horizontal factors be \(c\) and \(d\). So \(a\times c=6\), \(a\times d = 18\), \(b\times c=10\), \(b\times d=\)?

From \(a\times c = 6\) and \(a\times d=18\), divide the second equation by the first: \(\frac{a\times d}{a\times c}=\frac{18}{6}\Rightarrow\frac{d}{c}=3\Rightarrow d = 3c\).

From \(a\times c=6\) and \(b\times c = 10\), divide the second by the first: \(\frac{b\times c}{a\times c}=\frac{10}{6}=\frac{5}{3}\Rightarrow b=\frac{5}{3}a\).

Now, from \(a\times c = 6\), possible integer values for \(a\) and \(c\): (1,6),(2,3),(3,2),(6,1).

Let's take \(a = 2\), \(c = 3\) (since GCD(6,10)=2, so \(a = 2\) is a common factor). Then \(d=3c = 9\)? No, wait \(a\times d=18\), if \(a = 2\), then \(d=\frac{18}{2}=9\)? But \(b\times c = 10\), \(c = 3\), then \(b=\frac{10}{3}\), not integer.

Take \(a = 3\), \(c = 2\). Then \(d=\frac{18}{3}=6\). \(b\times c=10\), \(c = 2\), so \(b = 5\). Then \(b\times d=5\times6 = 30\). Wait, let's check:

Vertical factors: \(a = 3\), \(b = 5\)

Horizontal factors: \(c = 2\), \(d = 6\)

Check \(a\times c=3\times2 = 6\) (correct), \(a\times d=3\times6 = 18\) (correct), \(b\times c=5\times2 = 10\) (correct), \(b\times d=5\times6 = 30\) (correct).

Alternatively, the common factor for 6 and 18 is 6? No, GCD(6,18)=6. Then \(c = 1\) (since \(6\times1 = 6\)), \(d=\frac{18}{6}=3\). Then for 10, \(b\times1 = 10\), so \(b = 10\), then \(b\times d=10\times3 = 30\). Wait, same result.

Or common factor for 6 and 10 is 2. So vertical factor \(a = 2\), then \(c=\frac{6}{2}=3\), \(b=\frac{10}{2}=5\). Then horizontal factor for 18: \(d=\frac{18}{3}=6\). Then \(b\times d=5\times6 = 30\).

Step1: Find common vertical factor of 6 and 10

The greatest common divisor of 6 and 10 is \(GCD(6,10)=2\). So the left - hand vertical factor is 2.

Step2: Find horizontal factor for 6

Divide 6 by 2: \(\frac{6}{2}=3\). So the top - hand horizontal factor is 3.

Step3: Find horizontal factor for 18

Divide 18 by 3: \(\frac{18}{3}=6\). So the right - hand horizontal factor is 6.

Step4: Find vertical factor for 10

Divide 10 by 2: \(\frac{10}{2}=5\). So the bottom - hand vertical factor is 5.

Step5: Find the missing cell

Multiply the bottom - hand vertical factor (5) and the right - hand horizontal factor (6): \(5\times6 = 30\).

Answer:

The missing number in the factor puzzle is 30. The completed grid (with factors) is:

Vertical factors: 2 (left), 5 (right)

Horizontal factors: 3 (top), 6 (bottom)

10 (2×5)	30 (5×6)

Wait, more accurately, the missing cell (bottom - right) is 30.