Guest Sep 13, 2017

#1**+2 **

**The highest common factor (HCF) of 140 and x is 20.**

**The lowest common multiple (LCM) of 140 and x is 420.**

**Find the value of x.**

The following formula reduces the problem of computing the least common multiple

to the problem of computing the greatest common divisor (GCD),

also known as the greatest common factor:

\( \operatorname{lcm}(a,b)=\frac{|a\cdot b|}{\operatorname{gcd}(a,b)}. \)

\(\begin{array}{|rcll|} \hline |a\cdot b| &=& \operatorname{lcm}(a,b) \cdot \operatorname{gcd}(a,b) \\\\ && \operatorname{gcd}(140,x) = 20 \\ && \operatorname{lcm}(140,x) = 420 \\\\ 140\cdot x &=& 420 \cdot 20 \\ x &=& \frac{420 \cdot 20}{140} \\ \mathbf{ x } & \mathbf{=} & \mathbf{60} \\ \hline \end{array}\)

heureka Sep 13, 2017

#2**+1 **

heureka answers is (of course!) correct; but, if you don't have that formula at hand, you might think about it this way:

Since the highest common factor of 140 and x is 20, x must contain 20 as a factor.

Since the lowest common multiple of 140 and x is 420: because 140 divides into 420 three times, x must contain a 3.

Since x contains both 20 and 3 as factors, x is 20 · 3 = 60.

geno3141 Sep 13, 2017