Question

Find the HCF and LCM of 612 and 1314 by using the prime factorization method.
(a) 12 and 678
(b) 18 and 44400
(c) 24 and 19866
(d) 18 and 44676

Answer 1

Hint: To find the HCF and LCM of two numbers we are to find the prime factors of the numbers. Then we should try to analyze the factors. In that way we get the common prime factors and thus we reach our desired solution.

Complete step by step solution:
We are given two numbers, 612 and 1314.
And we are to find their HCF and LCM using a prime factorization method.
Now, to find the factors, \[1314 = 2 \times 3 \times 3 \times 73\]
As,
\[\begin{array}{l}
2\underline {\left| {1314} \right.} \\
3\underline {\left| {657} \right.} \\
3\left| {\underline {219} } \right.\\
73
\end{array}\]
And again,\[612 = 2 \times 2 \times 3 \times 3 \times 17\]
As,
\[\begin{array}{l}
2\underline {\left| {612} \right.} \\
2\underline {\left| {306} \right.} \\
3\left| {\underline {153} } \right.\\
3\left| {\underline {51} } \right.\\
17
\end{array}\]
So, HCF of 1314 and 612 would be,\[2\times 3\times 3=18\], as it is the product of smallest power of each common prime factor in the numbers.
And, now, LCM of 1314 and 612 would be,\[2\times 2\times 3\times 3\times 17\times 73=44,676\], as it is the product of highest power of each common prime factor in the numbers.
Hence, the correct option is, (d) 18 and 44676.

Note: The highest common factor (H.C.F) or greatest common divisor (G.C.D) of two numbers is the largest positive integer that perfectly divides the two given numbers. And again, the least common multiple of two numbers is the “smallest non-zero common number” which is a multiple of both the numbers. Both terms can be defined with the full forms of their abbreviations.