Question

How many numbers divisible by 132:
264, 396, 462, 792, 968, 2178, 5184, 6336

• A. 4
• B. 5
• C. 6
• D. 7

Hint:

To check divisibility quickly, divide the number mentally by the divisor and see whether the result is a whole number.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
We need to find how many numbers in the given list are exactly divisible by \(132\).
Given numbers: \[ 264,\ 396,\ 462,\ 792,\ 968,\ 2178,\ 5184,\ 6336 \]

Step 2: Key Formula or Approach:
A number is divisible by \(132\) if: \[ \frac{\text{Number}}{132} \] gives a whole number (integer).

Step 3: Detailed Explanation:
Check each number one by one: \[ 264 \div 132 = 2 \] Divisible.
\[ 396 \div 132 = 3 \] Divisible.
\[ 462 \div 132 = 3.5 \] Not divisible.
\[ 792 \div 132 = 6 \] Divisible.
\[ 968 \div 132 \neq \text{integer} \] Not divisible.
\[ 2178 \div 132 = 16.5 \] Not divisible.
\[ 5184 \div 132 \approx 39.27 \] Not divisible.
\[ 6336 \div 132 = 48 \] Divisible.
Thus, divisible numbers are: \[ 264,\ 396,\ 792,\ 6336 \] Total numbers divisible by \(132\): \[ 4 \]

Step 4: Final Answer:
The required count is: \[ \boxed{4} \] Hence, the correct option is: \[ \boxed{\text{(A) 4}} \]

Answer 2

The question asks how many of the eight given numbers are exactly divisible by 132. Since \( 132 = 4 \times 3 \times 11 \), a number is divisible by 132 only if it is divisible by 4, by 3, and by 11 all at once. Checking each number against these three simpler tests is a quicker route to the same count as long division.

1. 264: Last two digits 64 are divisible by 4; digit sum \( 2+6+4=12 \) is divisible by 3; alternating digit sum \( 4-6+2=0 \) is divisible by 11. All three hold, so 264 is divisible by 132.
2. 396: Last two digits 96 divide by 4; digit sum \( 3+9+6=18 \) divides by 3; alternating digit sum \( 6-9+3=0 \) divides by 11. Divisible by 132.
3. 462: Last two digits 62 are not divisible by 4 (\( 62 \div 4 = 15.5 \)), so 462 already fails the test for 4 and cannot be divisible by 132.
4. 792: Last two digits 92 divide by 4; digit sum \( 7+9+2=18 \) divides by 3; alternating digit sum \( 2-9+7=0 \) divides by 11. Divisible by 132.
5. 968: Last two digits 68 divide by 4, and the alternating digit sum \( 8-6+9=11 \) divides by 11, but the digit sum \( 9+6+8=23 \) is not divisible by 3, so 968 fails and is not divisible by 132.
6. 2178: Last two digits 78 are not divisible by 4 (\( 78 \div 4 = 19.5 \)), so this number already fails the test for 4.
7. 5184: Last two digits 84 divide by 4; digit sum \( 5+1+8+4=18 \) divides by 3; but the alternating digit sum \( 4-8+1-5=-8 \) is not divisible by 11, so 5184 is not divisible by 132.
8. 6336: Last two digits 36 divide by 4; digit sum \( 6+3+3+6=18 \) divides by 3; alternating digit sum \( 6-3+3-6=0 \) divides by 11. Divisible by 132.

Only four numbers, 264, 396, 792, and 6336, pass all three divisibility checks together and are therefore divisible by 132.

Therefore, the correct answer is 4.