Question

M= 2^4 x 3^3
n= 2^3 x 3^2 x13

find the highest common factor of 3m and 2n

Answer 1

144

Explanation:

We know that any integer can be written as a product of prime numbers.

Then if two given numbers share a given amount of prime numbers, then the product of all of these shared prime numbers is the highest common factor between these two numbers:

Here we have:

m = 2⁴*3³

n = 2³*3²*13

We want to find the highest common multiple of 3*m and 2*n, let's write these numbers, and remember that:

a*aⁿ = a⁽ⁿ ⁺ ¹⁾

Then:

3*m = 3*( 2⁴*3³) = 2⁴*3⁴

2*n = 2*( 2³*3²*13) = 2⁴*3²*13

So both of these numbers have the factor 2 four times.

And both of these numbers have the factor 3 at least 2 times.

So we can rewrite:

3*m = 2⁴*3⁴ = (2⁴*3²)*3²

2*n = 2⁴*3²*13 = (2⁴*3²)*13

Then the highest common multiple is:

2⁴*3² = 16*9 = 144