How many of the following numbers are divisible by 132: 264, 396, 462, 792, 968, 2178, 5184, 6336?

Introduction / Context:
This question asks us to check several given numbers for divisibility by 132 and count how many satisfy the condition. Rather than performing full long division for each, we can use factorization of 132 and divisibility rules for its prime factors. This tests knowledge of prime factorization and how to combine basic divisibility tests in an efficient way.

Given Data / Assumptions:

- Numbers given are 264, 396, 462, 792, 968, 2178, 5184 and 6336. - We must determine how many of these are exactly divisible by 132. - 132 is a composite number with small prime factors.

Concept / Approach:
First factorize 132 to understand its divisibility structure. We have 132 = 2^2 * 3 * 11. Therefore, a number is divisible by 132 if and only if it is divisible by 4, by 3 and by 11 at the same time. We can either check these three divisibility rules for each number or directly divide by 132 if comfortable with mental arithmetic. After testing each number, we count those with zero remainder.

Step-by-Step Solution:
Step 1: Factor 132: 132 = 2 * 66 = 2 * 2 * 33 = 4 * 33 = 4 * 3 * 11 = 2^2 * 3 * 11. Step 2: Check 264. Compute 264 ÷ 132 = 2, so 264 is divisible by 132. Step 3: Check 396. Compute 132 * 3 = 396, so 396 is divisible by 132. Step 4: Check 462. 132 * 3 = 396 and 132 * 4 = 528, so 462 lies between and is not a multiple of 132. Step 5: Check 792. 132 * 6 = 792, so 792 is divisible by 132. Step 6: Check 968. 132 * 7 = 924 and 132 * 8 = 1056, so 968 is not a multiple of 132. Step 7: Check 2178. 132 * 16 = 2112 and 132 * 17 = 2244, so 2178 is not a multiple of 132. Step 8: Check 5184. 132 * 39 = 5148 and 132 * 40 = 5280, so 5184 is not a multiple of 132. Step 9: Check 6336. 132 * 48 = 6336, so 6336 is divisible by 132. Step 10: The numbers divisible by 132 are 264, 396, 792 and 6336, a total of four numbers.

Verification / Alternative check:
As an alternative, we can check that each of these four numbers is divisible by 4, 3 and 11. For example, 264 is even and its last two digits 64 form a number divisible by 4, its digit sum 2 + 6 + 4 = 12 is divisible by 3, and the alternating sum for 11 is (2 + 4) − 6 = 0, a multiple of 11. Similar checks confirm that 396, 792 and 6336 satisfy all three divisibility rules, strengthening our conclusion.

Why Other Options Are Wrong:
Options B, C and D assume that more than four numbers are divisible by 132. However, we have shown that 462, 968, 2178 and 5184 fail the multiple test for 132, so they cannot be counted. Any answer larger than four overestimates the count.

Common Pitfalls:
Some learners may wrongly assume that being divisible by 4 and 3 is enough and forget the factor 11. Others may miscalculate products of 132 or rely on approximate multiplication that leads to off by one errors. Using clear multiplication steps or systematic divisibility rules helps avoid such mistakes.

Final Answer:
The number of given values that are divisible by 132 is 4, which matches option A.