1234 left behind

Number Theory Level 2

If the integer N N leaves a remainder of 1234 when divided by 2013, what is the remainder when N N is divided by 183 183 ?


The answer is 136.

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Arron Kau by Arron Kau

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7 solutions

A Joshi
Aug 18, 2013

The number N can be written in the format :

Dividend = Divisor * Quotient + Remainder as

N = 2013 * Q + 1234

This can be written as

N = 183 * 11 * Q + 1234

Dividing both sides by 183 , we get

N/183 = 11 * Q + 1234/183

N/183 = 11 * Q + (1098 +136)/183

N/183 = 11 * Q + 6 + 136/183

Multiplying both sides by 183 , we get

N = 183 * ( 11* Q + 6 ) + 136

If 11* Q + 6 = K

N = 183 * K + 136

which means 136 is the remainder when N is divided by 183

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Moderator note:

I've edited this solution to make the statements appear as separate lines. You should check the preview to ensure that your solution is presented as intended. Proper formatting helps makes your solution more readable.

I had added 1234 into 2013 = 3247 then divided 3247 by 183 and got 136

Pirah Sikandar - 7 years, 9 months ago

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I did the same process... :)

Joter Belleza - 7 years, 9 months ago

Its a really simple problem in reality but the thing is...u have 2 no how to solve it. :D

Siddharth Balakrishnan - 7 years, 9 months ago

sir great logic

Ramesh Khinchi - 7 years, 9 months ago

Take N=1234. N%183 = 136. Therefore, 136 is the answer.

3 Helpful 0 Interesting 0 Brilliant 0 Confused

Easier. (2013*n+1234)/183 = 136

Philip John - 7 years, 9 months ago

This seems to be true for all N that are congruent to 1234 mod 2013. Why is that the case?

Connor Stack - 7 years, 9 months ago

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2013 is divisible by 183...

Ryan Wood - 7 years, 9 months ago
Tamoghna Banerjee
Aug 19, 2013

183 is a factor of 2013. Moreover, N is exactly divisible by 2013. It follows that 183 completely divides N leaving no remainder. Now, 183 is a smaller number compared to 2013. Thus, it is clear that 183 can be added a number of times in order to get closest to 1234(since 183 leaves a remainder when it divides 1234.) Now, 183 when added upon it six times, it gets summed up to 1098 and after that it exceeds if we try to add another 183 over the sum. Thus 1098 is the closest number to 1234 which is divisible by 183. Hence the difference(i.e. 136) is the remainder.

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Moderator note:

You should be careful that others can understand what you are writing. Definitions that you use should be clear, and notation should conform with that given in the problem (where possible).

I would disagree with the claim that "N is exactly divisible by 2013", as this contradicts the first statement of the problem. Are you thinking of some other value?

I apologize for the blunder made... It will be (N-1234) which is exactly divisible by 2013. Moreover as 183 is also a factor of 2013 it will completely divide (N-1234) leaving no remainder. But 183 being smaller than 2013 can divide the term, N, further, diminishing the remainder of 1234 by an amount x(as it is clear 183 cannot completely divide 1234). Hence, it will be a term (1234-x) which will be the remainder when N is divided by 183. It comes out that when 183 is multiplied by 6 the product is 1098 and when multiplied by 7 the product is 1281(greater than the term 1234 and of course when added to N-1234 yields a number greater than N which is not at par with the question.) Thus Subtracting 1098 from 1234 we yield a term 136 which is finally the remainder(i.e. can no more be divided by 183.)

Master, Kindly comment on this explanation and oblige. I will be so very grateful to you for pointing out my misunderstanding of the problem.

Tamoghna Banerjee - 7 years, 9 months ago

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Yes, replacing N N with N − 1234 N-1234 will work.

The rest of the proof can be expressed clearer, but I understand what you are trying to say.

Calvin Lin Staff - 7 years, 9 months ago

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Thank you Master, I will certainly strive for a clearer way the next time.

Tamoghna Banerjee - 7 years, 9 months ago
Deepankar Singh
Aug 25, 2013

The number N can be written in the format :

Dividend = Divisor * Quotient + Remainder as

N = 2013 * Q + 1234

This can be written as

N = 183 * 11 * Q + 1234

Dividing both sides by 183 , we get

N/183 = 11 * Q + 1234/183

N/183 = 11 * Q + (1098 +136)/183

N/183 = 11 * Q + 6 + 136/183

Multiplying both sides by 183 , we get

N = 183 * ( 11* Q + 6 ) + 136

If 11* Q + 6 = K

N = 183 * K + 136

which means 136 is the remainder when N is divided by 183

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Urfi Ramadhani
Aug 20, 2013

N Mod 2013 = 1234,

N mod 183 = a,

N = 2013 + 1234 = 3247,

N mod 183 = 3247 Mod 183 = 136,

a = 136,

0 Helpful 0 Interesting 0 Brilliant 0 Confused
Ronald Salim
Aug 19, 2013

N=2013a+1234=3.11.61.a+1234

N/183=1234/183=6.7431693989071038251366120218579

6.7431693989071038251366120218579-6= 0.7431693989071038251366120218579.183=136

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Nelvson Shine
Aug 18, 2013

"If the integer N leaves a reminder of 1234 when divided by 2013"

since brilliant mathematics answers are integer values from 0 to 999 inclusive, so logically we can assume that N is 1234. N divided 2013 leaves 2013.

Now we find out what is the remainder when N is divided by 183, so :

= 1234 = 1234 mod 183 183

= 136 = 136

Thus it means the correct answer is 136

0 Helpful 0 Interesting 0 Brilliant 0 Confused

That doesn't matter. 2013 is divisible by 183, this means that we know that we only need to deal with 1234. N could be any multiple of 2013. But yes, the rest of it is right.

Ryan Wood - 7 years, 9 months ago

wew

Ronald Salim - 7 years, 9 months ago

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