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Generalization of a concept

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by verma.kumarrishikesh » Sun Sep 21, 2014 10:56 am
I have a query
we have a positive number X when divided by positive integer D gives remainder R.
Now we have a factor of D say m
when X is divided by m what would be the remainder?

one thing i have checked through number trials that it may not be R but we get the constant remainder and it can be determined. but would request for a logic behind this?
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Source: — Data Sufficiency |

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by [email protected] » Sun Sep 21, 2014 11:13 am
Hi verma.kumarrishikesh,

I think that there's an error in what you've written. The remainder would depend on the value of M, so the remainder is not necessarily a constant.

For example...
X = 30
D = 8

30/8 = 3r6

Factors of D = 1, 2, 4 , 8
So, ANY of these values could be the "M" that you mentioned.

Now, dividing X by the possible values of M would give us...
30/1 = 30r0
30/2 = 15r0
30/4 = 7r2
30/8 = 3r6

These 4 options produce a variety of remainders.

I think that you're attempting to draw a conclusion based on a practice question (or questions) that you've seen. If you'd like some extra information on those practice questions, then you should post them here.

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by verma.kumarrishikesh » Sun Sep 21, 2014 11:44 am
Dear Rich
I believe my post is not clear my apology, yes the same is aftermath of a question in the forum so i extended the same to see what i get.

the problem is like this

i am given a remainder and the divisor but not the actual dividend.
will i be able to calculate a remainder if a get a new divisor which is a factor of earlier divisor.

so in the illustration i get a divisor 8 and remainder 6 and now i have to find out what is the remainder when the same no is divided by 4.
I would get a 2 every time no matter what is the dividend.
i could not get the concept why this is happening and when i can get the same remainder 6.
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by Matt@VeritasPrep » Mon Sep 22, 2014 9:32 am
The question you're asking is one that belongs to the field of modular arithmetic: if you're really curious about this topic, here's a fantastic introduction. (If you're not that curious, here's a much shorter introduction.) I love number theory, so I highly recommend the topic.

To answer your specific question, let's consider a somewhat general example.

Suppose I give you the divisor 15 and the remainder 3. The quotient is unknown, so we'll call it q. This tells us that the dividend is of the form 15q + 3.

Now I want you to tell me the remainder when that same dividend is divided by 5 (a factor of the original divisor). The dividend is 15q + 3, and the divisor is 5. (15q + 3)/5, in the remainder system, gives 3q + 3, so the remainder remains the same.

Now suppose that I screwed up! The divisor was 15, but the remainder was actually 11. This time, the dividend is of the form 15q + 11. When we go to find the remainder when divided by 5, we'll get (15q + 11)/5, which gives us 3*(q+2) + 1, or a remainder of 1. Uh oh!

As we can see, the remainders are not consistent. I'd be happy to explain why, but delving much further into the topic would demand introducing a concepts irrelevant to the GMAT, however, so if you're curious I'd check it out after you finish this test!
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by [email protected] » Mon Sep 22, 2014 12:08 pm
Hi verma.kumarrishikesh,

As I noted in my earlier post, there is no "catch all" rule for what you're describing.

It sound like you tried a few examples, but there was a "bias" in how you chose your numbers, so the results appeared to be consistent. When you do a bit more work, you'll see that while there can be patterns when you limit the types of numbers you use, there is no constant remainder.

Using your example (divisor of 8, remainder of 6), then divisor of 4 instead:

22/8 = 2r6
22/4 = 5r2

30/8 = 3r6
30/4 = 7r2
Etc.

Here, we have a limited pattern because all the values are EVEN and 4 is exactly HALF of 8.

But what if we kept the divisor of 8 and the remainder of 6, but later used a divisor of 2 instead:

22/8 = 2r6
22/2 = 11r0

30/8 = 2r6
30/2 = 15r0

Now we have a DIFFERENT limited pattern.

So, unfortunately there is no pattern/rule that fits what you're looking for. "Remainder" questions aren't too common on the GMAT (you'll probably see 1-2), but most can be beaten by TESTing VALUES based on whatever information the prompt gives you to work with.

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