Question

Find the L.C.M. and H.C.F. of 70 and 80?

Answer 1

Hint: The least common multiple is the smallest number that is divided by the given two or more numbers exactly, while the highest common factor is the largest number that divides the given two or more numbers (HCF). We calculate the required object using this definition.

Complete step-by-step solution:
We express each number as a product of prime numbers to obtain the LCM of the given numbers. The LCM is the product of the highest power of the prime numbers that appear in prime factorization of any number.
We express each number as a product of prime numbers to obtain the HCF of the provided numbers. HCF is the prime factor with the highest value.
We have two numbers 70 and 80
We will expand 70 as product of prime numbers
$70 = 2 \times 5 \times 7$..................(1)
Now, we will expand 80 as product of prime factors
$80 = 2 \times 2 \times 2 \times 2 \times 5$
$80 = {2^4} \times 5$........................(2)
We can conclude from equation 1 and 2 that we have common factors as one 2 and one 5 while the extra factor is one 7 in 70 and three 2’s in 80.
So, the L.C.M. of 70 and 80 is
$ = 2 \times 5 \times 7 \times 2 \times 2 \times 2$
$ = 560$
And, the H.C.F. of 70 and 80 is
$ = 2 \times 5$
$ = 10$
Hence, the L.C.M. and H.C.F. of 70 and 80 is 560 and 10 respectively.\[{\text{Product of two numbers = }}\left( {L.C.M} \right) \times \left( {H.C.F} \right)\]

Note: The basic definitions of LCM and HCF are crucial in addressing this problem. So, we got our answer by applying the definition. We can use a relationship to check the accuracy of our answer:
\[{\text{Product of two numbers = }}\left( {L.C.M} \right) \times \left( {H.C.F} \right)\]