Question

There are 4 numbers. The HCF of each pair is 7 and the LCM of all the numbers is 1470. What is the product of the 4 numbers?
A.504210
B.502410
C.504120
D.501420

Answer 1

Hint: Here, we will use the formula to find the LCM of the four numbers without taking the highest common factor of the four numbers equals to the product of the four numbers and HCF, that is, \[{\text{LCM}} = {\text{Product of four numbers}} \times {\text{HCF}}\].
Apply this formula, and then use the given conditions to find the required value.

Complete step by step Answer :

Given that the HCF of each pair is 7 and the LCM of all the four numbers is 1470.

Let us assume that the four numbers are \[7a\], \[7b\], \[7c\] and \[7d\].

We know that the formula to find the value of LCM of the four numbers without taking the highest common factor of the four numbers is equals to the product of the four numbers and HCF, that is, \[{\text{LCM}} = {\text{Product of four numbers}} \times {\text{HCF}}\].

Substituting the values of HCF, LCM and the four numbers without HCF in the above formula for finding the LCM, we get

\[ \Rightarrow 1470 = abcd \times 7\]

Dividing the above equation by 7 on each of the sides, we get

\[
   \Rightarrow \dfrac{{1470}}{7} = \dfrac{{abcd}}{7} \times 7 \\
   \Rightarrow 210 = abcd \\
   \Rightarrow abcd = 210 \\
\]

We will now find the product of the four numbers.

\[
  {\text{Product of the numbers}} = 7a \times 7b \times 7c \times 7d \\
   = {7^4} \times abcd \\
\]

Substituting the above value of \[abcd\] in the above equation, we get

\[
   \Rightarrow {7^4} \times 210 \\
   \Rightarrow 504210 \\
\]


Thus, the product of the four numbers is 504210.

Hence, option C will be correct.

Note: In this question, we have to use the formula of the product of four numbers without taking the highest common factor and HCF is equals to LCM. Then use the given conditions and values given in the question, and substitute the values of HCF and LCM in the given formula for LCM, to find the required value. Also, we are supposed to write the values properly to avoid any miscalculation.