SOLUTION: the lcm of 84 and the n is 588 and their gcf is 28. What is n?

Question 933548: the lcm of 84 and the n is 588 and their gcf is 28. What is n?
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
the first number is 84
the second number is not known.
the least common multiple is 588
the greatest common factor is 28.

84 has to be equal to 28 * 3 because 84 / 28 = 3 and 28 is a factor of 84.

so we know that 84 = 28 * 3.

since 588 is a multiple of 84, then 588 / 84 = 7

so we know that 84 * 7 = 588 which simplifies to 28 * 3 * 7 = 588

since 588 is a multiple of our second number, and since 28 is a factor of our our second number, then we know that 28 * x = 588.

solve for x and we get x = 21

we get 28 * 21 = 588.

this can be simplified to 28 * 3 * 7 = 588.

our first number gets us 28 * 3 * 7 = 588

our second number gets us 28 * 3 * 7 = 588

our first number is 28 * 3 = 84

our second number must be 28 * 7 = 196.

our first numeber is 84.
our second number is 196.

the greatest common factor is 28 because 28 * 3 = 84 and 28 * 7 = 96.

the least common multiple is 588 because 84 * 7 = 588 and 196 * 3 = 588.

breaking both numbers down to their prime factors, we get:

84 = 4*7*3

196 = 4*7*7

common factors are 4*7 = 28

uncommon factors are 3 and 7 that are left after you strip out 4*7 from each.

least common multiple is therefore 3 * 196 and 7 * 84.

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