Question

Find the HCF and LCM of 8624 and 21658 by the fundamental theorem of arithmetic.
(a) HCF = 90, LCM = 1905914
(b) HCF = 98, LCM = 1905904
(c) HCF = 98, LCM = 1915904
(d) HCF = 88, LCM = 1905904

Answer 1

Hint: For solving this question we will first express the given numbers in terms of multiplication of prime numbers and then find the HCF (Highest common factor) and LCM (Least common multiple) of the given numbers.

Complete step-by-step answer:
Given:
Two numbers 8624 and 21658. We have to find their HCF and LCM by the fundamental theorem of arithmetic in which first we express any number in terms of multiplication of prime numbers. Then, we will find the HCF and LCM.
We can write, $8624=2\times 2\times 2\times 2\times 7\times 7\times 11$ and $21658=2\times 7\times 7\times 13\times 17$ .
Now, HCF of 8624 and 21658 can be determined by taking common prime factors of both numbers= $2\times 7\times 7$ = 98.
Now, as $8624= 2\times 2\times 2\times 2\times 7\times 7\times 11$ and $21658= 2\times 7\times 7\times 13\times 17$ . Then, LCM of 8624 and 21658 can be determined by taking the highest power of every prime number in the given numbers. Then,
$\begin{align}
  & \text{As, }8624= {{2}^{4}}\times {{7}^{2}}\times 11\text{ and }21658= 2\times {{7}^{2}}\times 13\times 17\text{ }\text{.} \\
 & \text{LCM = } {{2}^{4}}\times {{7}^{2}}\times 11\times 13\times 17=1905904\text{ }\text{.} \\
\end{align}$
Thus, LCM of 8624 and 21658 is 1905904.
Hence, (b) is the correct option.

Note: The question is very easy to solve but the student should not miss any factor while writing the factors of the given numbers and avoid doing calculation mistakes while solving the question. After we have calculated the HCF and LCM we should check whether it’s correct or not while doing simple arithmetic calculations.