Question

Find the greatest common factor of 126 and 3528

Answer 1

We have to find the greatest common factor of 126 and 3528.

To do that, we have to factorize each number.

This means that, for each number, we start dividing by 2. If the result of the quotient is still divisible by two, we divide it again by two, repeating until the number is not divisible by 2 anymore.

Then, we do the same with 3, as we have done with 2. And the same with all the prime factors until the quotient result is 1. Then we end the factorization.

With the two numbers factorized, we compare the factors in common and calculate the greatest common factor.

We have for 126 (each step is a division by the last number in the row) :

`\begin{gathered} 126...2 \\ 63\ldots3 \\ 21\ldots3 \\ 7\ldots7 \\ 1 \end{gathered}`

We have skipped 5 because 7 is not divisible by 5.

So we can write 126 as:

`126=2\cdot3\cdot3\cdot7=2\cdot3^2\cdot7`

The number 3528 can be written as:

`\begin{gathered} 3528\ldots2 \\ 1764\ldots2 \\ 882\ldots2 \\ 441\ldots3 \\ 147\ldots3 \\ 49\ldots7 \\ 7\ldots7 \\ 1 \end{gathered}`
`3528=2^3\cdot3^2\cdot7^2`

Then, we pick the common factors:

- We have 2, with the maximum common exponent of 1. So it is 2.

- We have 3, with a maximum common exponent of 2. So it is 3^2=9.

- And finlly, we have 7, with a maximum common exponent of 1. So it is 7.

We multiply all this factors and we get:

`2\cdot3^2\cdot7=2\cdot9\cdot7=126`

The greatest common factor of 126 and 3528 is 126.

NOTE: If we have two numbers, A and B, with A>B, and we have to look for the greatest common factor, the first check is to do A/B. If the remainder of the quotient is 0, then the greatest common factor is B, as in this case.