Find the greatest number which divides 285 and 1249 leaving remainders 9 and 7 respectively.

Given: The numbers 285 and 1249.

To find: The greatest number which leaves remainder 9 and 7 in 285 and 1249.

Concept Used:

Euclid's division lemma:

If there are two positive integers a and b,

then there exist unique integers q and r such that,

a = bq + r where 0 ≤ r ≤ b.

Explanation:

The new numbers after subtracting remainders are:

285-9 = 276

1249-7 = 1242

∴ The required number = HCF of 276 and 1242

Let a = 1242 and b = 276

As 1242>276

By applying Euclid’s division lemma,

1242 = 276q + r, (0≤r<276)

On dividing 1242 we get quotient as 4 and remainder r as 138.

⇒ 1242 = 276× 4 + 138

Now apply Euclid’s division lemma on 276 and 138,

276 = 138q+ r, (0≤r<138)

On dividing 276 we get quotient as 2 and remainder r as 0.

⇒ 276 = 138 × 2 + 0

∴ HCF = 138

Hence remainder is = 0

Hence required number is 138

Hence the greatest number which divides 285 and 1249 leaving remainder 9 and 7 respectively is 138.