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The largest number which is a factor of 66 and 110 is.
A.22
B.2
C.11
D.None of these

seo-qna
Last updated date: 29th Mar 2024
Total views: 400.2k
Views today: 11.00k
MVSAT 2024
Answer
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Hint: Focus on the largest common factor of two numbers that 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 answer:
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 66. We know 66 is an even number, so it can be written as $66=2\times 33$ . Further we can break 33 as $33=3\times 11$ . Therefore, we can write 66 as $2\times 3\times 11$ .
Now let us move to the factorisation of 110. So, as 110 is an even number, we can write it as $110=2\times 55$ . Again, 55 is an odd number, so 55 can be further written as: $55=5\times 11$ . So, finally we can write 110 in terms of its prime factor as: $110=2\times 5\times 11$ .
Now to find the H.C.F., we need to multiply all the common prime factors of the two numbers. So, HCF(66,110) is a product of one 2 and one 11.
\[HCF\left( 66,110 \right)=2\times 11=22\]
Therefore, we can conclude that the answer to the above question is option (a).

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 $HCF=\dfrac{HCF\text{ of numerator of the fractions}}{\text{LCM of the denominator of the fractions}}$ .