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# Find the value of x.

Sep 13, 2017

#1
+22188
+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:

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

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

Sep 13, 2017
#2
+17772
+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.

Sep 13, 2017