Find the largest number that divides 320 and 457 and leaves remainders 5 and 7 respectively.

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answer is B.

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Detailed Solution

Given are two numbers, 320 and 457, which when divided by a number leaves remainder 5 and 7 respectively.
For 320 and 457 to be exactly divisible by the largest number, the given remainders must be zero. So, we will subtract the remainder 5 from 320 and the remainder 7 from 457.
320 – 5 = 315
457 – 7 = 450
Now, the largest number will be the HCF of 315 and 450.
So, we proceed with the prime factorization of 315 and 450, to get their HCF as follows,

315 = 3 × 3 × 5 × 7 450 = 2 × 3 × 3 × 5 × 5

So, the product of the common factors among the two numbers is

3 × 3 × 5 = 45

.
Therefore, HCF (315, 450) = 45.
So, the largest number which divides 320 and 457 leaving the remainders 5 and 7 respectively is 45.
Hence, the correct option is 2.

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