Question

How many of the following numbers are divisible by132?
264, 396, 462, 792, 968, 2178, 5184, 6336
A. 6
B. 4
C. 5
D. 8

Answer 1

Hint: Take the prime factorization of 132. Compare its common factors to the common factor of the 8 numbers formed by prime factorization. Compare all the numbers to 132 and find the number which is divisible by 132.

Complete step-by-step answer:
We have been given 8 numbers, out of which we need to find the numbers that are divisible by 132. We can find it easily by finding the prime factorization of each number and comparing them to the prime factorization of 132. If they both have common multiples then that particular number is divisible by 132.
First let us find the prime factorization of 132. We know that prime factorization is finding the prime factor which multiplies together to make the original number, where the prime number is a number greater than 1.The only factors of prime numbers are 1 and the number itself.
\[132=4\times 3\times 11\], prime factorization of 132.
So if the number is divisible by all three numbers 4, 3 and 11, then the number is divisible by 132 also.
Now let us find the prime factorization of all 8 numbers.
\[\begin{align}
  & 264=2\times 2\times 2\times 3\times 11=2\times 4\times 3\times 11 \\
 & 396=2\times 2\times 3\times 3\times 11=4\times 3\times 11\times 3 \\
 & 462=2\times 11\times 3\times 7 \\
 & 792=2\times 2\times 2\times 3\times 3\times 11=2\times 3\times 4\times 3\times 11 \\
 & 968=2\times 2\times 2\times 11\times 11 \\
 & 2178=2\times 3\times 3\times 11\times 11 \\
 & 5184=2\times 2\times 2\times 2\times 2\times 2\times 3\times 3\times 3\times 3 \\
 & 6336=2\times 2\times 2\times 2\times 3\times 11\times 2\times 3=8\times 3\times 4\times 3\times 11 \\
\end{align}\]
Thus we found the prime factorization of all the numbers. The numbers 264, 396, 792, and 6336 are divisible by the three numbers 3, 4 and 11. Thus these three numbers are also divisible by 132.
Hence, these 4 numbers, 264, 396, 792, 6336 are divisible by 132.
Hence option B is the correct answer.

Note: You might be tempted to take the division of all the numbers by 132, which is also correct. But it takes time and prime factorization is easier to do than checking the divisibility by dividing all the numbers by 132. Or for the first few numbers you can also check with the multiplication table for 132.