Question

Find the largest no. That divides 2053 and 967 and leaves a remainder of 6 and 7 respectively

Answer 1

Find the largest number that divides 2053 and 967 and leaves a remainder as 5 and 7 respectively.

2053 leaves remainder 5

967 leaves remainder 7

→ 2053 - 5 = 2048

→ 967 - 7 = 960

The required number will be the HCF of 2048 and 960,

2048 = 2¹¹ × 1 = 2^6 × 2^5 × 1

960 = 2^6 × 3 × 5 × 1

HCF (2048,960) = 2^6

= 2×2×2×2×2×2

= 64

Therefore, the largest number that divides 2053 and 967 and leaves a remainder as 5 and 7 respectively is 64.

There's an alternate method too:

Let m be the required number.

Now, on dividing 2053 and 967 by m let the quotients be q1 and q2 respectively,

so, by Euclid 's division lemma,

2053 = mq1 + 5 ---- (i)

967  = mq2 + 7 ---- (ii)

Now, mq1 = 2048 and, mq2 = 960

clearly, hcf of mq1 and mq2 is m

so, m = H.C.F {2048, 960} = 64

Hope This Helps :)