Divisibility

[Troika]

Is the sum of integers a and b divisible by 7?

1. a is not divisible by 7.
2. a - b is divisible by 7.

OA: C

Source: MGMAT NP, q. 10, pg. 133

[Anurag@Gurome]

Is the sum of integers a and b divisible by 7?

1. a is not divisible by 7.
2. a - b is divisible by 7.

Statement 1: (a + b) may or may not be divisible by 7. For example, say,

• a = 8 and b = 1 --> (a + b) = 9 is not divisible by 7
  a = 7 and b = 0 --> (a + b) = 7 divisible by 7

Not Sufficient

Statement 2: (a + b) may or may not be divisible by 7.
Same examples as above.

Not Sufficient

1 & 2 Together: As a is not divisible by 7 but (a - b) is, b is also not divisible by 7. Hence, (a + b) is not divisible by 7.

Sufficient

The correct answer is C.

[Ashujain]

]
1 & 2 Together: As a is not divisible by 7 but (a - b) is, b is also not divisible by 7. Hence, (a + b) is not divisible by 7.

Sufficient

The correct answer is C.

Hi Anurag, can you please elaborate this point.

[Anurag@Gurome]

...As a is not divisible by 7 but (a - b) is, b is also not divisible by 7. Hence, (a + b) is not divisible by 7...

Hi Anurag, can you please elaborate this point.

We know that a is not divisible by 7 but (a - b) is.
Say, (a - b) = 7m, where m is any non-negative integer.

Now, let us assume b is divisible 7 too and b = 7n, where m is any non-negative integer.

Now, a = (a - b) + b = 7m + 7n = 7(m + n) --> Divisible by 7 --> Not possible

Hence, b is not divisible by 7

Hope that helps.

[Ashujain]

...As a is not divisible by 7 but (a - b) is, b is also not divisible by 7. Hence, (a + b) is not divisible by 7...

Hi Anurag, can you please elaborate this point.

We know that a is not divisible by 7 but (a - b) is.
Say, (a - b) = 7m, where m is any non-negative integer.

Now, let us assume b is divisible 7 too and b = 7n, where m is any non-negative integer.

Now, a = (a - b) + b = 7m + 7n = 7(m + n) --> Divisible by 7 --> Not possible

Hence, b is not divisible by 7

Hope that helps.

That's great! But can i generalize it as below:
If a and b are not divisible by n and a-b is divisible by n then a+b is not divisible by n?

[Anurag@Gurome]

...can i generalize it as below:
If a and b are not divisible by n and a-b is divisible by n then a+b is not divisible by n?

Yes, you can as long as n is an odd integer.

[Ashujain]

...can i generalize it as below:
If a and b are not divisible by n and a-b is divisible by n then a+b is not divisible by n?

If a and b are not divisible by n, then (a + b) is not divisible by n regardless of what (a - b) is.

Lets take following case:
a=10 b=4 n=7
a is not divisible by n and b is not divisible by n but a+b=14 is divisible by n=7

[Anurag@Gurome]

...
Lets take following case:

Thanks for pointing it out.
Edited the post.

[ankita1709]

...
Lets take following case:

Thanks for pointing it out.
Edited the post.

Then whats the answer?
It should be E I guess then

[Anurag@Gurome]

Then whats the answer?
It should be E I guess then

The correct answer is C as described in my first reply.

Let's go into a bit deep analysis.

From statement 1, we know that a is not divisible by 7. Let us assume a = (7m + k), where m is any non-negative integer and k is any positive integer less than 7. This is just the mathematical way to represent any integer which is not divisible by 7. For example, 5 can be written as (7*0 + 5), 16 can be written as (7*2 + 2) etc.

Now from statement 2, we know that (a - b) is divisible by 7. Let us assume (a - b) = 7n, where n is any non-negative integer.

Now, b = a - (a - b) = (7m + k) - 7n = 7(m - n) + k = (Some multiple of 7) + k
As k is any positive integer less than 7, b is not divisible by 7.

Now, (a + b) = (7m + k) + 7(m - n) + k = 7(2m - n) + 2k = (Some multiple of 7) + 2k

Now according to our assumption, k is any positive integer less than 7. Hence, possible values of k are 1, 2, 3, 4, 5, and 6. Hence, possible values of 2k are 2, 4, 6, 8, 10, and 12. None of this values when added to a multiple of 7 will result a multiple of 7.

Hence, (a + b) is not multiple of 7.

Hope this clears your doubt.

[Ashujain]

...
Lets take following case:

Thanks for pointing it out.
Edited the post.

Hi Anurag,

I guess we can generalize it for below case:
If a and b are not divisible by n and a-b is divisible by n then a+b is not divisible by n.
The above is true only when n is any odd integer. if n is even then we can not be sure. let me explain it as below:
a-b=nk whers k is any positive integer
then a+b=nk+2b so if n is odd then 2b is not divisible by n but if n is even then 2b may or may not be divisible by n.

Kindly correct me if I am wrong.

[Anurag@Gurome]

...If a and b are not divisible by n and a-b is divisible by n then a+b is not divisible by n.
The above is true only when n is any odd integer. if n is even then we can not be sure. let me explain it as below:
a-b=nk whers k is any positive integer
then a+b=nk+2b so if n is odd then 2b is not divisible by n but if n is even then 2b may or may not be divisible by n.

Kindly correct me if I am wrong.

You are absolutely correct.
For example, take a = 15 and b = 3.
Then, neither of a or b is divisible by 6 but both (a - b) = 12 and (a + b) = 18 are.