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# How many positive integers less than 100 have a remainder of

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Manager
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Re: How many positive integers less than 100 have a remainder of [#permalink]  21 Sep 2009, 20:42
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$$\frac{X-2}{13} = n$$ where n is a non-negative integer, and X is a positive integer less than 100. Basically determine how big the subset for possible values of n, and you have your answer.

Rearrange the above equation to:

13n + 2 = X

and since X < 100,

13n + 2 < 100
13n < 98
n < 98/13

Since n must be an integer....

n < 8

So there are 8 possible values for n (0 to 7), and therefore 8 positive integers less than 100 that have a remainder of 2 when divided by 13.
Senior Manager
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Re: How many positive integers less than 100 have a remainder of [#permalink]  21 Sep 2009, 21:42
My working was like AKProdigy87's
Except that 0 is not considered positive (or negative) so the answer should be 7.
Manager
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Re: How many positive integers less than 100 have a remainder of [#permalink]  21 Sep 2009, 23:41
[quote="pejmanjohn"]How many positive integers less than 100 have a remainder of 2 when divided by 13?

Easy one.
Number closest to 100 but less than 100 and divisible by 13 = 91.
91 = 13 * 7.
91+2 < 100.

Satisfies all the conditions.
Ans : 7. What is the OA?
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Re: How many positive integers less than 100 have a remainder of [#permalink]  22 Sep 2009, 11:47
I say 8

13n + 2 < 100

There are 7 possibilities that fit this, but that was not counting 2 (2 / 13 = 0 remainder 2)

So you have 8 positive integers with remainder 2 when divided by 13.... 2, 15, 28, 41, 54, 67, 80, 93
Manager
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Re: How many positive integers less than 100 have a remainder of [#permalink]  22 Sep 2009, 12:36
7

0 divided by anything is a 0 - not counted
1st to count is 15

We get
15+13x<100
13x<85
x<6

Plus 15 => 7 integers
Senior Manager
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Re: How many positive integers less than 100 have a remainder of [#permalink]  22 Sep 2009, 17:20
Actually 8 might be the right answer because the question stem is asking what are “the positive integers with a remainder”. This could be interpreted as referring to the dividend, in this case 2 is a valid dividend as 2/13 = 0r2. If this is the case the answer is 8. What is the OA?
Manager
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Re: How many positive integers less than 100 have a remainder of [#permalink]  24 Sep 2009, 10:54
How many positive integers less than 100 have a remainder of 2 when divided by 13?

Simple question.
Let the number be A
A = 13 * q + 2
here q can take values 0,1,...

since A has to be a two digit number, thus the max value of q that satisfies this condition is
q = 7
A = 13 * 7 + 2 = 93

Thus the number of positive integers is from 0 through to 7 inclusive
total of 8 numbers
Manager
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Re: How many positive integers less than 100 have a remainder of [#permalink]  21 Oct 2009, 20:26
1
KUDOS
0 is even number but is not a positive or negative number, so the sol should be 7
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Re: How many positive integers less than 100 have a remainder of [#permalink]  21 Oct 2009, 21:50
AKProdigy87 wrote:
$$\frac{X-2}{13} = n$$ where n is a non-negative integer, and X is a positive integer less than 100. Basically determine how big the subset for possible values of n, and you have your answer.

Rearrange the above equation to:

13n + 2 = X

and since X < 100,

13n + 2 < 100
13n < 98
n < 98/13

Since n must be an integer....

n < 8

So there are 8 possible values for n (0 to 7), and therefore 8 positive integers less than 100 that have a remainder of 2 when divided by 13.

Two things:

i. n is a non-negative integer
ii. X is a positive integer less than 100

If so, n has to be 8 including 0.
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Re: How many positive integers less than 100 have a remainder of [#permalink]  03 Jul 2011, 00:46
I dont understand how 2 can be included in the set.

2 div by 13

13) 2 ( -> 13) 20 (0. -> 13) 20 (0.1

Where do you get a reminder of 13?

Thanks.
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Re: How many positive integers less than 100 have a remainder of [#permalink]  11 Oct 2011, 08:49
+1 for 8.

similar to the previous explanation.
x is positive & n is just a "non negative number" which means '0' is ok. So 0 through 7 = total 8.
Re: How many positive integers less than 100 have a remainder of   [#permalink] 11 Oct 2011, 08:49
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