Find the greatest number that will divide 445, 572 and 699 leaving remainder 4,5,6 effectively

By BYJU'S Exam Prep

Updated on: September 25th, 2023

63 is the greatest number that will divide 445, 572, and 699 leaving the remainder 4,5,6 effectively. We can have two explanations for the answer we have derived. We have to first subtract the remainder with the numbers. Then we can either use the prime factorization method or Euclid’s dilemma method to get the HCF which will help us get our answer. Detailed solutions via both methods are provided below.

Table of content

1. Greatest number that Divides 445, 572 and 699 with Remainders 4,5,6
2. Using Euclid’s Dilemma
3. Find the greatest number that will divide 445, 572 and 699 leaving remainder 4,5,6 effectively

Greatest number that Divides 445, 572 and 699 with Remainders 4,5,6

To find the greatest number that will divide 445, 572 and 699 leaving the remainder 4,5,6 effectively, we will follow the steps mentioned below.

we know the remainder of 445 is 4, 572 is 5, and 699 is 6. We will now subtract the numbers with the remainder –

445 – 4 = 441
572 – 5 = 567
699 – 6 = 693

Now, we will find the highest common factors of these newly derived numbers –

441 = 3 x 3 x 7 x 7
572 = 3 x 3 x 3 x 3 x 7
693 = 3 x 3 x 7 x 11

For all these numbers the common factors are the following

3 x 3 x 7 = 63

Now, we know that the HCF (441,567,693) = 63

Hence, 63 is the greatest number that will divide 445, 572 and 699 leaving remainder 4,5,6 effectively.

Using Euclid’s Dilemma

Another way to solve this question is by employing Euclid’s dilemma. As per it, a = bq + r where 0 ≤ r < b.

We will first, repeat the same step of subtracting remainders from the number which we did above. After it, we will use the following equations –

567 = 441 × 1 + 126

Assuming 441 as divisor and 126 as the remainder 126, we will derive the equation –

441 = 126 × 3 + 63

Repeating the process above, taking 126 as divisor and the remainder 63, we will derive –

126 = 63 × 2 + 0

As we can see, now the remainder equals 0. which cannot be replaced further. Thus, the answer we get is 63.

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