Hcf of two numbers is 4 & their lcm is 224. if one number is 32, the other number is,

HCF of two numbers is the highest common number, which is available in both the numbers. Before we proceed ahead to find the HCF, let us discuss what HCF is. HCF or highest common factor is the factor of any two or more numbers, which are common among them. Sometimes, it is also called the greatest common factor (GCF) or greatest common divisor (GCD).

Table of Contents Show

HCF of two numbers By Prime Factorisation
HCF of two numbers by Division Method
HCF of Three Numbers
Video Lesson on Numbers

For example, the HCF of 2 and 4 is 2, because 2 is the number which is common to both 2 and 4. For such small numbers, finding HCF is an easy method. But for larger numbers, we need to use different techniques such as prime factorisation and long division method, to find the HCF.

Let us discuss both the techniques.

HCF of two numbers By Prime Factorisation

Let us solve some examples here to understand this method.

Question 1: What is the HCF of 24 and 36?

Solution: By prime factorisation, we can write the two numbers;

24 = 2 x 2 x 2 x 3

36 = 2 x 2 x 3 x 3

Hence, after factoring the numbers 24 and 36, we can see, the factors 2x2x3 are common.

Therefore, the HCF (24, 36) = 2x2x3 = 12

Question 2: What is the HCF of 35 and 55?

Solution: By prime factorisaton we can write the two numbers as:

35 = 5 x 7

55 = 5 x 11

Hence, we can see the highest common factor for 35 and 55 here is 5.

Therefore, HCF (35, 55) = 5

Also, read:

Prime Factorization Of Hcf And Lcm
Properties Of Hcf And Lcm
Hcf And Lcm

HCF of two numbers by Division Method

We have already understood the prime factorisation method to determine the HCF.

Steps for Division method:

If we were given two numbers, then

First, divide the large number by a small number.
If the remainder is left, then divide the first divisor by remainder.
If the remainder divides the first divisor completely, then it is the HCF or highest common factor of the given two numbers.
If the remainder does not divide the first divisor completely, then repeat the steps.

Let us now learn the division method with the help of examples

Question:1 What is the HCF of 120 and 100.

Solution: Now, let us find the HCF by using division method.

Divide 120 by 100.

120/100 → 1 and remainder is 20

Now, divide the first divisor 100 by first remainder 20.

100/20 → 5 and remainder is 0.

Therefore, 20 is the HCF of 120 and 100.

Question 2: Find the HCF of 45 and 60 by the division method.

Solution: Divide 60 by 45.

60/45 → 1 and remainder is 15

Now, divide 45 by 15

45/15 → 3

Therefore, 15 is the HCF of 45 and 60.

HCF of Three Numbers

Let us solve an example when we need to find the HCF of three numbers.

Example: Find the HCF of 126, 162 and 180.

Solution: By prime factorisation, we can write the given numbers as;

126 = 2x3x3x7

162 = 2x3x3x3x3

180 = 2x2x3x3x5

Taking out the common factors of 126, 162 an d180, we get:

HCF(126,162,180) = 2x3x3 = 18

Video Lesson on Numbers

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Answer

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Hint: Here we will assume the two numbers to be any variables and then we will multiply them with the HCF. Then we will use the relation between the HCF and LCM of two numbers. From there, we will get all the possible pairs of numbers.

Complete step-by-step answer:

It is given that the LCM of the two numbers is equal to 400 and the HCF of the numbers is equal to 4.Since HCF of the numbers is 4. So, two numbers are multiples of 4.Let the numbers be \[4x\] and \[4y\].Now, we will use the relation between the HCF and LCM of two numbers.We know that the product of the given natural numbers is always equal to the product of HCF and LCM of the given natural numbers.Using this relation, we get\[4x \times 4y = 4 \times 400\]Multiplying the terms on both sides, we get\[ \Rightarrow 16xy = 1600\]Now, we will divide both sides by 16. Therefore, we get \[ \Rightarrow \dfrac{{16xy}}{{16}} = \dfrac{{1600}}{{16}}\]On further simplification, we get\[ \Rightarrow xy = 100\]The two numbers i.e. \[4x\] and \[4y\] are equal to 4 only when \[x\] and \[y\] are co prime numbers.So the possible pairs such that \[x\] and \[y\] are co prime numbers will be:-\[ \Rightarrow \left( {x,y} \right) = \left( {25,4} \right);\left( {100,1} \right)\]So the numbers will be \[\left( {4 \times 25,4 \times 4} \right) = \left( {100,16} \right)\] and \[\left( {4 \times 100,4 \times 1} \right) = \left( {400,4} \right)\].

Hence, there are only two possible pairs of the two numbers.

Note: To solve this question, we need to know the meaning of LCM and HCF and also the relation between them. LCM stands for the Lowest Common Factor and is defined as the smallest number that is a multiple of all these numbers. HCF stands for the Highest Common Factor and HCF of the given numbers is defined as the largest number that divides all these numbers completely. When the HCF of two integers or numbers is 1 then they are known as co prime numbers.