Question

Find the G.C.D. of 120, 504 and 882.

Answer 1

- Hint: Focus that G.C.D. of numbers is equal to the HCF of the numbers. To determine the HCF of the numbers, express the number in terms of the product of its prime factors and multiply all the prime common factors among all the numbers.

Complete step-by-step solution -

Before proceeding with the solution, let’s understand the concept of prime factorization. A prime number is a number which is not divisible by any other number except 1 and itself. Any number can be expressed as a product of prime numbers. All the prime numbers, which when multiplied, give a product equal to a number (say x) are called the prime factors of the number x. To write the prime factors of a number, we should always start with the smallest prime number, i.e. 2 and check divisibility. If the number is divisible by the prime number, then we write the number as a product of the prime number and another number, which will be the quotient when the given number is divided by the prime number. Then, we take the quotient and repeat the same process. This process is repeated till we are left with 1 as the quotient.
For example: Consider the number 51. It is an odd number. So, it is not divisible by 2. The sum of the digits of 51 is 5 + 1 = 6. Hence, 51 is divisible by 3. Now, $51=3\times 17$ . Now, we take 17. We know, 17 is a prime number. Hence, the prime factors of 51 are 3 and 17.
Now starting with finding the factors of 84. We know 84 is an even number, so it can be written as $2=2\times 42$ . Further we can break 42 as $42=2\times 7\times 3$ . Therefore, we can write 84 as $2\times 2\times 3\times 7$ .
Now let us move to the factorisation of 504. So, as 504 is an even number, we can write it as $504=2\times 252$ . Again, 252 is also an even number, so 252 can be further written as: $252=2\times 126$ . 126 also being even can be written as $126=2\times 63$ . And we can write 63 as: $63=3\times 3\times 7$ . So, finally we can write 504 in terms of its prime factor as: $504=2\times 2\times 2\times 3\times 3\times 7$ .
Now we will find the factors of 120. We know 120 is divisible by 12 and 10. Also, we know a number divisible by 10 must have 2 and 5 as its prime factors. Also, being divisible by 12, makes two 2s and 3 as its factors. So, 120 can be written as $120=2\times 2\times 2\times 3\times 5$ .
Finally we will move to find the prime factors of 882. So, as 882 is even, it can be written as $882=2\times 441$ . 441 is a multiple of 3, so we can write 441 as $441=3\times 147$ . Again 147 is divisible by 3 and gives the quotient 49 when divided by 3 and 49 is the square of 7. So, as a whole 882 can be written as: $882=2\times 3\times 3\times 7\times 7$ .
Now to find the G.C.D., we need to multiply all the common prime factors of the three numbers. So, GCD(120,504,882) is a product of one 2, zero 5s, zero 7s and one 3.
\[GCD\left( 120,504,882 \right)=2\times 3=6\]
Therefore, we can conclude that the G.C.D. of 120, 504 and 882 is 6.

Note: Be careful while finding the prime factors of each number. Also, it is prescribed that you learn the division method of finding the HCF as well, as it might be helpful. If in case you are asked to find the HCF of two fractions you must use the formula $HCF=\dfrac{HCF\text{ of numerator of the fractions}}{\text{LCM of the denominator of the fractions}}$ .