To get the Least Common Multiple (LCM) of 37 and 12 we need to factor each value first and then we choose all the factors which appear in any column and multiply them:

37: 3712: 223 LCM: 22337The Least Common Multiple (LCM) is: 2 x 2 x 3 x 37 = 444

4 0

3 years ago

The GCD(a, b) = 9, the LCM(a, b)=378. Find the least possible value of a+b.

denis-greek [22]

$\mathrm{gcd}(a,b)=9\implies9\mid a\text{ and }9\mid b\implies9\mid a+b$

which means there is some integer $k$ for which $a+b=9k$ .

Because $9\mid a$ and $9\mid b$ , there are integers $n_1,n_2$ such that $a=9n_1$ and $b=9n_2$ , and

$\mathrm{lcm}(a,b)=\mathrm{lcm}(9n_1,9n_2)=9\mathrm{lcm}(n_1,n_2)=378\implies\mathrm{lcm}(n_1,n_2)=42$

We have $42=2\cdot3\cdot7$ , which means there are four possible choices of $n_1,n_2$ :

1, 42
2, 21
3, 14
6, 7

which is to say there are also four corresponding choices for $a,b$ :

9, 378
18, 189
27, 126
54, 63

whose sums are:

387
207
153
117

So the least possible value of $a+b$ is 117.

6 0

3 years ago

Answer? ............​

Answer? ............