Find greatest numbers that will divide …

We have to  find the greatest number that divides 445, 572 and 699 and leaves remainders of 4, 5 and 6 respectively. This means when the number divides 445, 572 and 699 leaves remainders 4, 5 and 6 is that
445 - 4 = 441, 572 - 5 = 567 and 699 - 6 = 693 are completely divisible by the  required number.For the highest number which divides the above numbers can be calculated by HCF .
Therefore, the required number  is the  H.C.F. of 441, 567 and 693 Respectively.
First, consider 441 and 567.
By applying Euclid’s division lemma, we get
567 = 441 {tex}\times{/tex} 1 + 126
441 = 126 {tex}\times{/tex} 3 + 63
126 = 63 {tex}\times{/tex} 2 + 0.
Therefore, H.C.F. of 441 and 567 = 63
Now, consider 63 and 693
again we have to  apply Euclid’s division lemma, we get
693 = 63 {tex}\times{/tex} 11 + 0.
Therefore, H.C.F. of 441, 567 and 693  is  63
Hence, the required number is 63. 63 is the highest number which divides 445,572 and 699 will leave 4,5 and 6 as remainder respectively.