Question

suppose that the greatest common divisor of the positive integers X and Y is 21 and the least common multiple of X and Y is 252. The sum of all possible values of X is:

a) 20 b) 28 c)420 d)588

Answer 1

the greatest common divisor is the same as the greatest common factor btw.

Answer 2

so we need to look at
\[\gcd(x,y)=21 ; lcm(x,y)=\frac{x \cdot y}{\gcd(x,y)}=\frac{x \cdot y}{21} =252 \]
\[x \cdot y =252(21)\]

Answer 3

so we can play with these x's and y's right here

Answer 4

so here are two examples of what x can be 252,21

remember we need x>21

Answer 5

126,42

Answer 6

63,84

Answer 7

i think that is it x can be
252,21,126,42,63,84

Answer 8

so the sum of these is 588

Answer 9

all of those pairs i wrote have gcd 21

Answer 10

guess we should also check to see if they have lcm 252

Answer 11

we know the first pair does

Answer 12

the second pair does not
126=126*1
126=42*3
lcm(42,126)=126
so we should throw that pair out

Answer 13

so we have 252,21,63,84
sum is 420
but what is the lcm of 63 and 84...

Answer 14

ok lcm(63,84) is 252 my bad
so possible x's are 252,21,63,84
the sum of those is 420