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# How many integers from 0 to 50, inclusive, have a remainder

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How many integers from 0 to 50, inclusive, have a remainder [#permalink]

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24 Mar 2009, 23:51
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How many integers from 0 to 50, inclusive, have a remainder of 1 when divided by 3 ?

A. 15
B. 16
C. 17
D. 18
E. 19
[Reveal] Spoiler: OA

Last edited by Bunuel on 06 Nov 2012, 03:52, edited 2 times in total.
Renamed the topic and edited the question.

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Re: PS: 0 to 50 inclusive, remainder [#permalink]

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25 Mar 2009, 00:16
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C:17

I brute forced this one.
All the multiples of 3 + 1 will have a remainder of 1:
4,7,10,13,16,19,22,25,28,31,34,37,40,43,46,49 - 16 numbers total
But then I thought, no way it's this easy and thought about 1. 1/3 would also have a remainder of 1, making the answer 17.

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Re: PS: 0 to 50 inclusive, remainder [#permalink]

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25 Mar 2009, 01:10
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The range of numbers that leave a remainder of 1:
1-49

Therefore total number of integers:
(49-1)/3 + 1 = 17

The last +1 is because the question is inclusive.

Hence C

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Re: PS: 0 to 50 inclusive, remainder [#permalink]

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25 Mar 2009, 01:21
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My ans is also C.17.

Explanation:

1 also gives 1 remainder when divided by 3, another number is 4, then 7 and so on.
Hence we have an arithmetic progression: 1, 4, 7, 10,..... 49, which are in the form 3n+1.
Now we have to find out number of terms.
tn=a+(n-1)d, where tn is the nth term of an AP, a is the first term and d is the common difference.
so, 49 = 1+(n-1)3
or, (n-1)3 = 48
or, n-1 = 16
or, n = 17

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15 May 2010, 22:01
bibha wrote:
How many integers from 0 to 50, inclusive, have a remainder of 1 when divided by 3?
A.14 B.15. C.16 D.17 E.18

Ans: C
y=x*q+n
a1=1*3+1=4, a2=2*3+1=7, => d=a2-a1=3
an-I know that 51/17=3, in order to have a reminder +1, i will take x=16
an=16*3+1=49

N=[(an-a1)/d]+1= (49-4/3)+1=16

p.s. I may be wrong, if q=0 also could be used than a1=1 and not 4, and than N=17, Ans D.

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Last edited by PTK on 16 May 2010, 01:02, edited 2 times in total.

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16 May 2010, 02:42
bibha wrote:
Hey pkit,

an-I know that 51/17=3, in order to have a reminder +1, i will take x=16
an=16*3+1=49

Could you please explain this in detail??

I have meant that 51=3*x, than x=17, since we have limits from 0 to 50 inclusive the last number in sequence is 49 which is =16*3+1, as 50=16*3+2.
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17 May 2010, 09:02
well Bibha,
I would like to expalin this in a logical manner rather than by mathematical expression.
each no. which is 1 more than multiple of three would be in this series. so upto 50 there are 16 such no.s so correspondingly there are 16 no.s which are 1 more than a multiple of three and the last being 49.
reply me if it is not clear.

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How many integers from 0 to 50, inclusive, have a remainder [#permalink]

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14 Jun 2010, 01:11
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How many integers from 0 to 50, inclusive, have a remainder of 1 when divided by 3 ?

A. 15
B. 16
C. 17
D. 18
E. 19

Algebraic way:

Integer have a remainder of 1 when divided by 3 --> $$n=3p+1$$, where $$p$$ is an integer $$\geq{0}$$, so $$n$$ can take the following values: 1, 4, 7, ...

$$n=3p+1\leq{50}$$ --> $$3p\leq{49}$$ --> $$p\leq{16\frac{1}{3}}$$ --> so $$p$$, can take 17 values from 0 to 16, inclusive.

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14 Jun 2010, 21:53
Does it mean that if zero is included in any similar problem like this then we should consider it, no matter what the divisor is???
Ex- if question say any no. between 0 and 50, inclusive, divisible by 3
Answer will still be 17 Is it right
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14 Jun 2010, 22:05
hardnstrong wrote:
Does it mean that if zero is included in any similar problem like this then we should consider it, no matter what the divisor is???
Ex- if question say any no. between 0 and 50, inclusive, divisible by 3
Answer will still be 17 Is it right

0 is a multiple of every integer (except zero itself), so there are $$\frac{48-0}{3}+1=17$$ numbers divisible by 3 in the range 0-50 inclusive (check this: totally-basic-94862.html?highlight=range).

But in original question 0 is not considered as one of the numbers: the lowest value of n is 1 (for p=0) and the highest value of n is 49 (for p=16), so total of 17 such numbers.
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14 Jun 2010, 22:40
Good to know something new every time i login on this forum
Thanks
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15 Jun 2010, 11:54
bibha wrote:
How many integers from 0 to 50, inclusive, have a remainder of 1 when divided by 3?
A.14 B.15. C.16 D.17 E.18

((Last - First)/ n) +1 -> (49-1)/3 +1 = 17

We use 49 because that is the last that will produce a remainder of 1 when divided by 3 and 1/3 has a remainder of 1.

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Re: PS: 0 to 50 inclusive, remainder [#permalink]

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22 Jul 2010, 21:13

There are 48/3=16 numbers that are divisible by 3 (0 is excluded). If we add 1 to each and every of these number, we still have 16 numbers that have a remainder of 1 when divided by 3 (say 4, 7, 10,..., 49). But we have not counted 0 yet, 0+1 equals 1, 1 has a remainder of 1 when divided by 3.

Hence C.

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Re: PS: 0 to 50 inclusive, remainder [#permalink]

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31 Aug 2010, 23:56
I think we can find this very quickly by using this:
approximately 1/3 of the numbers from 0-50 are div by 3 so, (50/3) +1 = 16 + 1 = 17
approximately 1/2 of the numbers are div by 2 => (50/2 ) + 1 = 26
approximately 1/4 of the numbers are div by 4 => (50/4) + 1 =13
approximately 1/5 of the numbers are div by 5 => (50/5) + 1 =11
approximately 1/6 of the numbers are div by 6 => (50/6) + 1 = 9
I saw this on Bunuel's post somewhere and I thought to myself, how did he know? so I tried to calculate it and voila!, it is true...didn't try with range starting with non-zero numbers though
don't forget to add one..before you are done!

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Re: PS: 0 to 50 inclusive, remainder [#permalink]

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01 Sep 2010, 00:12
when 1 is devided by 3 remainder is 1, and when 3 is devided remainder 0.
16*3=48. so upto 48 there will be 16 number which will give remainder 1 after 48, there is only 1 number up to 50 which will give 1 remainder so the answer 17

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Re: PS: 0 to 50 inclusive, remainder [#permalink]

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06 Jan 2012, 06:22

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Re: PS: 0 to 50 inclusive, remainder [#permalink]

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06 Jan 2012, 20:57
First, lets look at the range. With 1 remainder, 4-49.

So, number of elements = (49-3)/3 + 1 = 16

Now, lets not forget 1, since 1/3 -> 1 as remainder (Good one!)

So, 16 +1 = 17
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Re: PS: 0 to 50 inclusive, remainder [#permalink]

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18 Oct 2012, 20:57
General Formula : n= px+q
p=3
q=1
n=3x+1
Substituting Values n=0,4,7...49 .
x = 0 to 16, inclusive
Hence Total nos : 17

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Re: How many integers from 0 to 50, inclusive, have a remainder [#permalink]

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04 Feb 2013, 23:48
Bunuel wrote:

Algebraic way:

Integer have a remainder of 1 when divided by 3 --> $$n=3p+1$$, where $$p$$ is an integer $$\geq{0}$$, so $$n$$ can take the following values: 1, 4, 7, ...

$$n=3p+1\leq{50}$$ --> $$3p\leq{49}$$ --> $$p\leq{16\frac{1}{3}}$$ --> so $$p$$, can take 17 values from 0 to 16, inclusive.

hi, i was wondering how we got 17 from the calculation that $$p\leq{16\frac{1}{3}}$$ ?

my approach was:

50-0+1 = 51 integers total

0/3 has r = 0
1/3 has r = 1
2/3 has r = 2

3/3 has r = 0
... and so on

thus there will be 1 value for every three that will have a remainder of 1 when divided by 3 (cyclicity of 3?)

so 51/3 = 17

would this method work for similar questions?

i was wondering how we treat the $$p\leq{16\frac{1}{3}}$$ term when trying the algebraic method? i.e. how do we know to arrive at 17 ?

thanks!

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Re: How many integers from 0 to 50, inclusive, have a remainder [#permalink]

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05 Feb 2013, 04:00
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essarr wrote:
Bunuel wrote:

Algebraic way:

Integer have a remainder of 1 when divided by 3 --> $$n=3p+1$$, where $$p$$ is an integer $$\geq{0}$$, so $$n$$ can take the following values: 1, 4, 7, ...

$$n=3p+1\leq{50}$$ --> $$3p\leq{49}$$ --> $$p\leq{16\frac{1}{3}}$$ --> so $$p$$, can take 17 values from 0 to 16, inclusive.

hi, i was wondering how we got 17 from the calculation that $$p\leq{16\frac{1}{3}}$$ ?

i was wondering how we treat the $$p\leq{16\frac{1}{3}}$$ term when trying the algebraic method? i.e. how do we know to arrive at 17 ?

thanks!

We have that $$n=3p+1$$, where $$p$$ is an integer $$\geq{0}$$. So, n can be 1 (for p=0), 4 (for p=1), 7, 10, ..., and 49 (for p=16) --> 17 values for p --> 17 values for n.

OR: $$p\leq{16\frac{1}{3}}$$ implies that p can take integer values from 0 to 16, inclusive, thus it can take total of 17 values.

Hope it's clear.
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Re: How many integers from 0 to 50, inclusive, have a remainder   [#permalink] 05 Feb 2013, 04:00

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