Answer :

**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.**

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