Question

How many of the following numbers are divisible by 132?
264, 396, 462, 792, 968, 2178, 5184, 6336
(a) 4
(b) 5
(c) 6
(d) 7

Answer 1

Hint: To find whether the given numbers are divisible by 132 or not, we will find the highest common factor of 132 with all the given numbers one by one. The case where the HCF comes out to be 132 we can say that 132 will divide that number or that number will be divisible by 132.

Complete step-by-step answer:
We know that the highest common factor of two numbers is the greatest number that divides both of them.
Here, we may use the Euclid’s division algorithm to find the highest common factor or the greatest common divisor of 132 with all the given numbers.
The Euclid’s division algorithm is the process of applying the Euclid’s Lemma in succession several times to obtain the greatest common divisor of any two numbers.
The last non-zero remainder that will be obtained in this process will be the greatest common divisor of the two numbers.

Coming to all the given numbers one by one:
For 264:
$264=132\times 2+0$
So, the greatest common divisor of 264 and 132 is 132.
Therefore, 264 is divisible by 132.
For 396:
$396=132\times 3+0$
So, the greatest common divisor of 132 and 396 is 132.

Therefore, 396 is also divisible by 132.
For 462:
$\begin{align}
  & 462=132\times 3+66 \\
 & 132=66\times 2+0 \\
\end{align}$

So, the greatest common divisor of 462 and 132 is 66.
Therefore, 462 is not divisible by 132.
For 968:
$\begin{align}
  & 968=132\times 7+44 \\
 & 132=44\times 3+0 \\
\end{align}$

So, the greatest common divisor of 968 and 132 is 44.
Therefore, 968 is not divisible by 132.
For 792:
$792=132\times 6+0$

So, the greatest common divisor of 792 and 132 is 132.
Therefore, 792 is divisible by 132.
For 2178:
$\begin{align}
  & 2178=132\times 16+66 \\
 & 132=66\times 2+0 \\
\end{align}$

So, the greatest common divisor of 2178 and 132 is 66.
Therefore, 2178 is not divisible by 132.
For 5184:
$\begin{align}
  & 5184=132\times 39+36 \\
 & 132=36\times 3+24 \\
 & 36=24\times 1+12 \\
 & 24=12\times 2+0 \\
\end{align}$

So, the greatest common divisor of 5184 and 132 is 12.
Therefore, 5184 is not divisible by 132.
For 6336:
$6336=132\times 48+0$

So, the greatest common divisor of 6336 and 132 is 132.
Therefore, 6336 is divisible by 132.
So, total 4 numbers from the given numbers are divisible by 132.
Hence, option (a) is the correct answer.

Note: While applying Euclid’s Division Lemma, if we get zero as the remainder in the first step, then the smaller of the two numbers becomes the greatest common divisor.