Divisors And Remainders

Number Theory Level 3

In dividing a number by 585, a student employed the method of short division. He divided the number successively by 5, 9 and 13 (factors 585) and got the remainders 4, 8, 12 respectively. If he had divided the number by 585, the remainder would have been $\text{\_\_\_\_\_\_\_\_\_\_\_}$ .

The answer is 584.

1 solution

Tran Quoc Dat
Mar 25, 2016

Let $A$ denote the number he used to divide by $585$ . $A$ divides by $5$ remains $4$ , divides by $9$ remains $8$ , divides by $13$ remains $12$ . So $A+1$ is divisible by all $5, 9, 13$ . Therefore, $A+1$ is divisible by $585$ , so $A$ divides $585$ remains $584$ .

Divisors And Remainders

The answer is 584.

1 solution

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