Question

grid with numbers 49, 42, 24 and a blank cell (implied problem: find the missing number in the grid, likely using factoring/multiplication relationships)

Explanation:

Step1: Find GCD of 49 and 42

First, find the greatest common divisor (GCD) of 49 and 42. The factors of 49 are 1, 7, 49; the factors of 42 are 1, 2, 3, 6, 7, 14, 21, 42. So GCD(49, 42) = 7.

Step2: Find the missing number

Let the missing number be \( x \). We can observe the relationship between the numbers in the same row or column. Looking at the second column: 42 and 24. Let's check the ratio or GCD. Wait, another approach: in a 2x2 grid, often the product of the diagonals is equal or there is a common factor. Wait, 49 is \( 7\times7 \), 42 is \( 7\times6 \), 24 is \( 6\times4 \)? Wait, no. Wait, 49 and 42 have a common factor of 7, 42 and 24 have a common factor of 6. Wait, maybe the first row first column is related to 49 and 24, and 42. Let's see: 49 divided by 7 is 7, 42 divided by 7 is 6, 24 divided by 6 is 4? No. Wait, another way: the number in the first row first column (let's call it \( x \)) and 49 are in the first column, 24 and 42 in the second column. Let's check the ratio of 42 to 24: \( \frac{42}{24}=\frac{7}{4} \)? No, \( \frac{42}{24}=\frac{7}{4} \) is wrong, \( \frac{42}{24}=\frac{7}{4} \) simplifies to \( \frac{7}{4} \)? Wait, 42 ÷ 6 = 7, 24 ÷ 6 = 4. 49 ÷ 7 = 7, so \( x \) ÷ 4 = 7? Then \( x = 28 \)? Wait, no. Wait, 49 and 42: GCD is 7, 42 and 24: GCD is 6. Wait, 7 and 6: maybe the missing number is 7×4=28? Wait, no. Wait, let's check the product of rows: 49 + 42 = 91, 24 + x =? No, product: 49×24 = 1176, 42×x = 1176? Then x = 1176 ÷ 42 = 28. Yes, that works. Because in a 2x2 grid, sometimes the product of the diagonals is equal (49×24 = 42×x). So:

$$ 49\times24 = 42\times x $$

$$ x=\frac{49\times24}{42} $$

Simplify: 49 and 42 have a common factor of 7, 24 and 42 have a common factor of 6. So \( \frac{49}{42}=\frac{7}{6} \), then \( x = \frac{7}{6}\times24 = 7\times4 = 28 \).

Answer:

28