Question

find the greatest number that divides 2733, 5619 and 6087 leaving a remainder of 3 in each case

Answer 1

Given data:

A number divides 2733, 5619, and 6087 leaving a remainder of 3 in each case.

To find:

The greatest number that justifies the given data.

Solution:

Given numbers are 2733, 5619, and 6087.

Let the greatest number that divides each of them leaving a remainder of 3 be 'x'

As it is said that the number divides them leaving 3 as remainder,

The numbers that are exactly divided by 'x' are:

2733 - 3 = 2730

5619 - 3 = 5616

6087 - 3 = 6084

Now, the HCF of the three numbers 2730, 5616, 6084 will be the greatest number to divide the numbers 2733, 5619, and 6087 leaving a remainder 3 in each case.

Finding HCF of 2730, 5616, 6084 by using Euclid's algorithm:

As 5616 > 2730, we apply the division lemma to 5616 and 2730:

5616 = 2730 x 2 + 156

As the remainder resulted in 2730 ≠ 0, we apply division lemma to 156 and 2730:

2730 = 156 x 17 + 78

Here, the new divisor is 156 and the new remainder is 78. On applying division lemma, we get:

156 = 78 x 2 + 0

There the remainder becomes zero, ending the procedure here.

The HCF of 2730 and 5616 is 78

Notice that 78 = HCF(156,78) = HCF(2730,156) = HCF(5616,2730) .

Now, let us see if 78 can manage to be the factor of 6084 too.

Since 6084 > 78, we apply the division lemma to 6084 and 78, to get

6084 = 78 x 78 + 0

As the remainder is zero, the HCF of 78 and 6084 is 78

Notice that 78 = HCF(6084,78).

So, altogether 78 is the HCF of 2730, 5616, 6084

Therefore, 78 is the greatest number that divides 2733, 5619, and 6087 leaving a remainder of 3 in each case.

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