Video: CBSE Class X • Pack 1 • 2018 • Question 15

CBSE Class X • Pack 1 • 2018 • Question 15

05:23

Video Transcript

Find the highest common factor and lowest common multiple of 404 and 96 and verify that the highest common factor timesed by the lowest common multiple is equal to the product of the two numbers.

We will start by finding the highest common factor. In order to do this, we will write both 404 and 96 as a product of primes. Let’s write a prime factor tree for 404.

We can see that 404 is two timesed by 202. And since two is prime, we can circle it. Now, 202 is simply two timesed by 101. And both two and 101 are prime. So we can circle both of those. And we’ve actually finished this prime factorization tree.

Now, let’s write the prime factorization tree for 96. We can see that 96 is two timesed by 48. And since two is prime, we circle it. And then 48 is four timesed by 12. Neither of which are prime. Now, we can divide four into two and two, both of which are prime. And 12 splits into four and three, where three is prime. And then, four splits into two and two and two is prime. So we can circle both of these.

Now, we found all the prime factors of 404 and 96. And we can write as a product of primes that 404 is equal to two squared timesed by 101 and 96 is equal to two to the five timesed by three.

Now, in order to find the highest common factor of these two numbers, we need to find the common prime factors and then the smallest exponent of the shared prime factors. So we can see that these two prime factorizations have a common factor of two. And the smallest exponent comes in the factorization of 404 with the exponent of two. And so therefore, the highest common factor is equal to two to the power of two or two squared. And two squared is simply equal to four.

So now, we have found that the highest common factor is four. Now, let’s find the lowest common multiple of 404 and 96. In order to do this, we will again be using the prime factorizations of 404 and 96. However, this time, we need to find all the different prime factors in the factorizations of 404 and 96 and their highest exponents. So for the different prime factors, we have two which appears in both the prime factorizations of 404 and 96, we have 101, and we also have three.

Since two appears in both factorizations, we will only be interested in the one with the largest exponent. So this is the two in the factorization of 96 since its exponent is five. And now, we have that the lowest common multiple is equal to the product of all the different prime factors with their largest exponents. And this gives us two to the power of five times three times 101.

When calculating this product, we can notice that two to the five times three is also equal to 96 since this is identical to the prime factorization of 96. And we obtain that our lowest common multiple is equal to 96 times 101. And now, we can write 101 as 100 plus one. And so we can see that the lowest common multiple is equal to 96 times 100 plus 96 times one, which is equal to 9600 plus 96. And so therefore, the lowest common multiple is 9696.

Now that we’ve found the highest common factor and the lowest common multiple, all that remains to do is to show that the highest common factor multiplied by the lowest common multiple is equal to the product of the two numbers. Using the two values that we’ve calculated, we can see that the highest common factor timesed by the lowest common multiple is equal to four times by 9696. And the product of the two numbers given in the question is 404 timesed by 96. And we can write 404 as four timesed by 101. So this product is also equal to four times 101 times 96.

When finding the lowest common multiple, we actually multiplied 101 by 96. And we got the answer to be 9696. And therefore, this product becomes four timesed by 9696. And this is quite clearly equal to the highest common factor timesed by the lowest common multiple.

So now, we have verified that the highest common factor multiplied by the lowest common multiple is equal to the product of the two numbers. Since we have found the highest common factor, which is four, the lowest common multiple, which is 9696, and we have verified this fact, we’ve now completed the question.

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