Question

Find the LCM and HCF of the following pairs of integers and verify that product of the two $$LCM{\text{ }} \times {\text{ }}HCF{\text{ }} = $$numbers. (i) 26 and 91 (ii) 510 and 92 (iii) 336 and 54

Answer 1

Hint: Here in this question, we have to find and verify the Highest common factor (HCF) and least common multiple (LCM) of given numbers by using a method of prime factorization. Prime factorization of a number means to express the given number as a product of the prime factors, that is, the product of the numbers that are prime numbers and divide the given number completely. We can find out the correct answer using this information.

Complete step-by-step answer:
The Least Common Multiple (LCM) of two numbers is the smallest number that has both of the first two numbers as factors.
The largest positive integer which divides two or more integers without any remainder is called
Highest Common Factor (HCF) or Greatest Common Divisor or Greatest Common Factor (GCF).

Prime factorization is the process of finding the prime numbers, which are multiplied together to get the original number.
A number that is not divisible by any other number except 1 and itself is known as a prime number. For example, 5 is not divisible by any number other than 1 and 5.
Consider the given numbers 26 and 91.
I.Let us find the prime factors of 26 and 91:
$$26 = 2 \times 13$$
$$91 = 7 \times 13$$
The highest common factor (HCF) of 26 and 91 is
$$\therefore $$ HCF of $$\left( {26,91} \right) = 13$$
and
The Least Common Multiple (LCM) of 26 and 91 is
LCM of $$\left( {26,91} \right) = 2 \times 7 \times 13$$
$$\therefore $$ LCM of $$\left( {26,91} \right) = 182$$
 To verify,
$$ \Rightarrow HCF\left( {26,91} \right) \times LCM\left( {26,91} \right) = 26 \times 91$$
$$ \Rightarrow \,\,13 \times 182 = 26 \times 91$$
 $$\therefore 2366 = 2366$$
Hence, verified.

II.Consider the given numbers 510 and 92.
Let us find the prime factors of 510 and 92:
$$510 = 2 \times 3 \times 5 \times 17$$
$$92 = 2 \times 2 \times 23$$
The highest common factor (HCF) of 510 and 92 is
$$\therefore $$ HCF of $$\left( {510,92} \right) = 2$$
and
The Least Common Multiple (LCM) of 510 and 92 is
LCM of $$\left( {510,92} \right) = 2 \times 2 \times 3 \times 5 \times 23 \times 17$$
$$\therefore $$ LCM of $$\left( {510,92} \right) = 23460$$
 To verify,
$$ \Rightarrow HCF\left( {510,92} \right) \times LCM\left( {510,92} \right) = 510 \times 92$$
$$ \Rightarrow \,\,2 \times 23460 = 510 \times 92$$
 $$\therefore 46920 = 46920$$
Hence, verified.

III.Consider the given numbers 336 and 54.
Let us find the prime factors of 336 and 54
$$336 = 2 \times 2 \times 2 \times 2 \times 3 \times 7$$
$$54 = 2 \times 3 \times 3 \times 3$$
The highest common factor (HCF) of 26 and 91 is
HCF of $$\left( {336,54} \right) = 2 \times 3$$
$$\therefore $$ HCF of $$\left( {336,54} \right) = 6$$
and
The Least Common Multiple (LCM) of 26 and 91 is
LCM of $$\left( {336,54} \right) = 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 3 \times 7$$
$$\therefore $$ LCM of $$\left( {336,54} \right) = 3024$$
 To verify,
$$ \Rightarrow HCF\left( {336,54} \right) \times LCM\left( {336,54} \right) = 336 \times 54$$
$$ \Rightarrow 6 \times 3024 = 336 \times 54$$
 $$\therefore 18144 = 18144$$
Hence, verified.

Note: By factoring a number we mean to express it as a product of two numbers or it can also be defined as the division of a given number by some other number such that the remainder is zero. Usually, negative factors of a number are not considered and the fractions can never be considered as a factor of a number. Factors of a number are defined as all the possible sets of two numbers whose product is equal to the given number. Thus, we should remember the difference between a factor and prime factor and solve a question by keeping the mentioned information in mind.