What is the greatest number that will divide 445 572 and 699 leaving remainder 4 5 and 6 respectively?

Text Solution

Solution : `445−4=441` <br> `572−5=567` <br> `699−6=693` <br> The greatest common factors of `441, 567 and 693 is,` <br> `441=3×3×7×7` <br> `567=3×3×3×3×7` <br> `693=3×3×7×11` <br> The common factors are `3×3×7=63.` <br> `445/63=7 with remainder as 4.` <br> `572/63=9 with remainder as 5.` <br> `699/63=11 with remainder as 6.` <br> Therefore, `63` is the greatest number.

Since the respective remainders of 445, 572, and 699 are 4, 5, and 6, we have to find the number which exactly divides (445-4), (572-5), and (696-6).So, the required number is the HCF of 441, 567, and 693.Firstly, we will find the HCF of 441 and 567.

∴ HCF = 63
Now, we will find the HCF of 63 and 693.

∴ HCF = 63
Hence, the required number is 63.

Given: 445, 572 and 699.

To find: Here we have to find the value of the greatest number which divides 445, 572 and 699 leaving remainders 4, 5 and 6 respectively.

Solution:

If the required number divide 445, 572 and 699 leaving remainders 4, 5 and 6 respectively, then this means that number will divide 441(445 $-$ 4), 567(572 $-$ 5) and 693(699 $-$ 6) completely.

Now, we just have to find the HCF of 441, 567 and 693.

First, let's find HCF of 441 and 567 using Euclid's division algorithm:

Using Euclid’s lemma to get: 

• $567\ =\ 441\ \times\ 1\ +\ 126$

Now, consider the divisor 441 and the remainder 126, and apply the division lemma to get:

• $441\ =\ 126\ \times\ 3\ +\ 63$

Now, consider the divisor 126 and the remainder 63, and apply the division lemma to get:

• $126\ =\ 63\ \times\ 2\ +\ 0$

The remainder has become zero, and we cannot proceed any further. 

Therefore the HCF of 441 and 567 is the divisor at this stage, i.e., 63.

Now, let's find HCF of 63 and 693 using Euclid's division algorithm:

Using Euclid’s lemma to get: 

• $693\ =\ 63\ \times\ 11\ +\ 0$

The remainder has become zero, and we cannot proceed any further. 

Therefore the HCF of 63 and 693 is the divisor at this stage, i.e., 63.

So, the greatest number which divides 445, 572 and 699 leaving remainders 4, 5 and 6 respectively is 63.