What is the value of x so that the seven digit number 5656x52 is divisible by 72?

Options

A. a) 4
B. b) 7
C. c) 5
D. d) 8

Answer

To find the value of \( x \) such that the seven-digit number \( 5656x52 \) is divisible by 72, we need to check the divisibility rules for both 8 and 9, since \( 72 = 8 \times 9 \).

### Step 1: Check for divisibility by 9
A number is divisible by 9 if the sum of its digits is divisible by 9.

The digits of the number \( 5656x52 \) are:
- 5, 6, 5, 6, \( x \), 5, 2

Calculating the sum:
\[ 5 + 6 + 5 + 6 + x + 5 + 2 = 29 + x \]

Now, we need \( 29 + x \) to be divisible by 9.

### Step 2: Find possible values of \( x \)
To find \( x \), we can check the values from 0 to 9:
- If \( x = 0 \): \( 29 + 0 = 29 \) (not divisible by 9)
- If \( x = 1 \): \( 29 + 1 = 30 \) (not divisible by 9)
- If \( x = 2 \): \( 29 + 2 = 31 \) (not divisible by 9)
- If \( x = 3 \): \( 29 + 3 = 32 \) (not divisible by 9)
- If \( x = 4 \): \( 29 + 4 = 33 \) (divisible by 9)
- If \( x = 5 \): \( 29 + 5 = 34 \) (not divisible by 9)
- If \( x = 6 \): \( 29 + 6 = 35 \) (not divisible by 9)
- If \( x = 7 \): \( 29 + 7 = 36 \) (divisible by 9)
- If \( x = 8 \): \( 29 + 8 = 37 \) (not divisible by 9)
- If \( x = 9 \): \( 29 + 9 = 38 \) (not divisible by 9)

The possible values of \( x \) that make the sum divisible by 9 are \( 4 \) and \( 7 \).

### Step 3: Check for divisibility by 8
A number is divisible by 8 if the last three digits form a number that is divisible by 8. The last three digits of our number are \( x52 \).

Now we will check the possible values of \( x \):
- If \( x = 4 \): \( 452 \div 8 = 56.5 \) (not divisible by 8)
- If \( x = 7 \): \( 752 \div 8 = 94 \) (divisible by 8)

### Conclusion
The only value of \( x \) that satisfies both conditions (divisible by 9 and divisible by 8) is \( x = 7 \).

Thus, the value of \( x \) is:
\[ \boxed{7} \]