Find the largest number dividing 230 and 142 and leaving remainders 5 and 7 respectively.

Let k be the largest number dividing 230 and 142 leaving remainders 5 and 7 respectively.

Therefore, we can write 230 = ka + 5 and 142 = kb + 7

Where, a, b

230 = ka + 5 142 = kb + 7

⇒ ka = 230 – 5 ⇒ kb = 142 – 7

ka = 225 kb = 135

But, k is the largest number.

So, k becomes the largest divisor, g. c. d of 225 and 135

225 = 5 × 5 × 3 × 3

135 = 5 × 3 × 3 × 3

Therefore, g. c. d(225, 135) = k

g. c. d(225, 135) = 5 × 3 × 3 = 45

k = 45

Therefore, the largest number dividing 230 and 142 leaving remainders 5 and 7 respectively is 45.