Question

Find the largest two digit number that divides 673 and 865, leaving remainder 1 in each.

Answer 1

Solution :-

To find the largest two digit number that divides 673 and 865, leaving remainder 1 in each case, first we have to subtract 1 from both the numbers and then we will compute H.C.F. of both the numbers.

⇒ 673 - 1 = 672
⇒ 865 - 1 = 864

H.C.F. of 672 and 864

              ___________
           2 |672, 864
              |___________
           2 |336, 432
              |___________
           2 |168, 216
              |___________
           2 | 84,  108  
              |___________
           2 | 42,    54 
              |___________
           3 |  21,   27
              |___________
              |    7,    9

H.C.F. of 672 and 864 = 2*2*2*2*2*3 = 96

So, 96 is the largest two digit number that 673 and 865, leaving remainder 1 in each case.

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Let us check our answer.

865 ÷ 96

Quotient = 9

Remainder = 1

673 ÷ 96

Quotient = 7

Remainder = 1

So, our answer is correct.

Answer 2

In order to make 673 and 865 completely divisible ,first we need to deduct the remainder from the number & then find the H.C.F of the numbers.

i.e.1 from both the cases

673-1=672
865-1=864

So, numbers completely divisible by are 672 and 864

672 = 2×2×2×2×2×3×7

864 = 2×2×2×2×2×3×3×3

H.C.F= 2×2×2×2×2×3=96

H.C.F of 672 & 864= 96

Hence, the largest no. which divides 673and 865 leaving remainder 1 in each case is 96

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Hope this will help you.