Time for brush up of number system

Number Theory Level 1

When 242 is divided by a certain divisor the remainder obtained is 8. When 698 is divided by the same divisor the remainder obtained is 9. However, when the sum of the two numbers 242 and 698 is divided by the divisor, the remainder obtained is 4. What is the value of the smallest divisor?

The answer is 13.

2 solutions

Sophie Crane
Apr 18, 2014

From the first statement, the divisor must divide $234$ exactly. From the second statement, it must divide $689$ exactly. The greatest common divisor of 234 and 689 is 13 (found by Euclid's algorithm: $689=2 \times 234+221, 234=221+13, 221=17 \times 13+0$ ). Check thirteen with the third statement, and it's fine.

If n is the divisor, we have 234=8(mod n) and 689=9(mod n) If S=234+689 we have S=17=4(mod n) then 13=0(mod n) and 13 is prime, then n=13

Ángela Flores - 7 years, 1 month ago

Rajnish Kaushik
Apr 24, 2014

Let the divisor be d.

When 242 is divided by the divisor, let the quotient be 'x' and we know that the remainder is 8.

Therefore, 242 = xd + 8

Similarly, let y be the quotient when 698 is divided by d.

Then, 698 = yd + 9.

242 + 698 = 940 = xd + yd + 8 + 9

940 = xd + yd + 17

As xd and yd are divisible by d, the remainder when 940 is divided by d should have been 17.

However, as the question states that the remainder is 4, it would be possible only when17 by d leaves a remainder of 4.

If the remainder obtained is 4 when 17 is divided by d, then d has to be 13.

Time for brush up of number system

The answer is 13.

2 solutions

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