NP 4th Edition Chapter 10, In Action Page 133 Problem 10

[georgepa]

Is the sume of integers a and b divisible by 7?
(1) a is not divisible by 7
(2) a - b is divisible by 7

I get how (1) and (2) alone by themselves are not sufficient

From the answer key:

Using the coinmbined values it "tells us that a and b have the SAME REMAINDER when divided by 7

I can't understand the above sentence from which the proof is shown. Can someone please explain that to me.

[2amitprakash]

This means that remainders for 'a' and 'b' when divided by 7 are same. This is a conclusion you can draw if 'a' is not divisible by 7 but a-b is.

for example, if a=15, then b must be 8 or 1 to be divisible by 7 (considering only positive numbers but it doesn't matter if you consider -ve too). So you can see that remainder in both case is same i.e. 1.

For me, this is sufficient to tell that a+b is not divisible by 7 and hence answer must be 'C'.

[georgepa]

This means that remainders for 'a' and 'b' when divided by 7 are same. This is a conclusion you can draw if 'a' is not divisible by 7 but a-b is.

I see the example and understand. I think the following is the mathematical proof for those who want to understand why.

(1) Let: a/7 = X + R1 = 7A + R1 => [X, the quotient is divisible by 7]

(2) Let (a-b)/7 = Y + 0 => [Divisibility by 7]

(3) a/7 -b/7 = Y = 7B => [Y, the quotient, is a multiple of 7 ]

(4) (7A + R1) - b/7 = 7B => [Substituting for a/7 from (1)]

(5) b/7 = (7A -7B) + R1

(6) b/7 = 7(A-B) + R1 => [A quotient and a remainder of a number b when divided by 7]

[esledge]

Nice proof, George and Amit. I'll just add one thing to this comment by Amit:

This means that remainders for 'a' and 'b' when divided by 7 are same. This is a conclusion you can draw if 'a' is not divisible by 7 but a-b is.

for example, if a=15, then b must be 8 or 1 to be divisible by 7 (considering only positive numbers but it doesn't matter if you consider -ve too). So you can see that remainder in both case is same i.e. 1.

If the remainder is the same for either a or b is divided by 7 (say, remainder = x, where x = 1,2,3,4,5,or6), then the remainder when a+b is divided by 7 is the sum of those individual remainders, 2x (or 2x-7, if 2x > 7). This remainder is a non-zero even number, which proves that there is a remainder.

This question would not be so straightforward if all of the 7's had been replaced with 6's. I think the answer would be E in that case...