Understanding Divisibility by 132

The question asks us to find out how many numbers in the given list are divisible by 132. The list of numbers is 660, 754, 924, 1452, 1526, 1980, 2045, and 2170.

To determine if a number is divisible by 132, we first need to understand the factors of 132. We can find the prime factorization of 132:

$$132 = 2 \times 66$$

$$132 = 2 \times 2 \times 33$$

$$132 = 2 \times 2 \times 3 \times 11$$

So, $132 = 4 \times 3 \times 11$. The numbers 4, 3, and 11 are pairwise coprime (they have no common factors other than 1). This means a number is divisible by 132 if and only if it is divisible by 4, 3, and 11 simultaneously.

Divisibility Rules for 3, 4, and 11

Let's recall the divisibility rules for 3, 4, and 11:

• Divisibility by 3: A number is divisible by 3 if the sum of its digits is divisible by 3.
• Divisibility by 4: A number is divisible by 4 if the number formed by its last two digits is divisible by 4.
• Divisibility by 11: A number is divisible by 11 if the difference between the sum of the digits at odd places (from the right) and the sum of the digits at even places (from the right) is either 0 or a multiple of 11.

Now, we will check each number in the list against these rules.

Checking Divisibility of Each Number by 132

We will go through the list of numbers one by one and check if they meet the divisibility criteria for 3, 4, and 11.

From the table above, we can see which numbers are divisible by 3, 4, and 11. A number must be divisible by all three (3, 4, and 11) to be divisible by 132.

Summary of Divisible Numbers

The numbers from the list that are divisible by 132 are:

• 660
• 924
• 1452
• 1980

Let's count how many such numbers we found. There are 4 numbers in the list that are divisible by 132.

Revision Table: Divisibility Concepts

Additional Information: General Divisibility Strategies

When checking divisibility by a composite number, finding its prime factorization is a very helpful first step. If the factors are pairwise coprime, you can check divisibility by each factor separately using their respective rules. For example:

• To check divisibility by 6, check divisibility by 2 and 3 (since $6 = 2 \times 3$, and 2 and 3 are coprime).
• To check divisibility by 10, check divisibility by 2 and 5 (since $10 = 2 \times 5$, and 2 and 5 are coprime).
• To check divisibility by 15, check divisibility by 3 and 5 (since $15 = 3 \times 5$, and 3 and 5 are coprime).

If the factors are not pairwise coprime (e.g., checking divisibility by 8 which is $2 \times 4$), you cannot just check divisibility by 2 and 4. You must check by the highest power of each prime factor in the factorization (e.g., for 8, check by $2^3=8$). In the case of 132 ($2^2 \times 3 \times 11$), the factors are 4 ($2^2$), 3, and 11, which are pairwise coprime, making the separate checks valid.