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

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Senior Manager
Joined: 15 Sep 2011
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Re: How many integers from 0 to 50, inclusive, have a remainder  [#permalink]

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01 Jul 2015, 16:55
Calculated the same. $$\frac{Last R1 - First R1}{3} +1 = \frac{(49 - 1)}{3} +1 = 16 + 1 = 17$$.

Testing the answer choices if another multiple is needed or if there are one too many. This proved that C was the correct answer.

A. 15 * 3 = 45. 50-45 = R5. Too low.
B. 16 * 3 = 48. 50-48 = R2.
C. 17 * 3 = 51. 50-51 = R1, which is in line with what the question asks.
D. 18 * 3 = 54. 54-50= R4. Too high
E. 19 * 3 = 57. 57-50= R7.Too high
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How many integers from 0 to 50, inclusive, have a remainder  [#permalink]

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14 Sep 2016, 09:47
Economist wrote:
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

First, when a integer is divided by 3, it can have the remainder 0,1 or 2.

There are 50-0+1=51 integers between 0 and 50 inclusive.

Just a quick example:
integer 0: remainder 0
integer 1: remainder 1
integer 2: remainder 2
integer 3: remainder 0
integer 4: remainder 1 and so on. Practicaly, there are 51/3=17 remainders of 0, 17 remainders of 1 and 17 remainders of 2. Answer: C
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Re: How many integers from 0 to 50, inclusive, have a remainder  [#permalink]

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14 Sep 2016, 10:11
Number of integers from 0 to 50 with remainder of 1 when divided by 3 can be found out as follows:
Number of integers divisible by 3 from 0 to 50: 48 = 3 + (n-1) * 3 ; using the formula for nth term of series in arithmetic progression ,where nth term is 48, 1st term is 3 and difference between terms is 3.
Therefore n=16, since adding 1 to each of these numbers will give remainder 1, however, when 1 is divided by 3, it gives remainder of 1, hence we have to include 1, giving the total number of integers as 16+1 =17
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How many integers from 0 to 50, inclusive, have a remainder  [#permalink]

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14 Sep 2016, 11:03
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

let x=number of integers with a remainder of 1 when divided by 3
range is 1-49
1+3(x-1)=49
x=17
C.
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Re: How many integers from 0 to 50, inclusive, have a remainder  [#permalink]

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26 Sep 2017, 16:24
Economist wrote:
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

The first number that has a remainder of 1 when divided by 3 is 1, and the last number is 49.

Thus, the number of integers from 0 to 50 inclusive that have a remainder of 1 when divided by 3 is:

(49 - 1)/3 + 1 = 17

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Posts: 1229
How many integers from 0 to 50, inclusive, have a remainder  [#permalink]

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01 Apr 2018, 06:04
abhishekik wrote:
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

hello there

from this $$49 = 1+(n-1)3$$ I get following:

$$49 = 1+3n-3$$

$$49=3n-2$$

$$3n=51$$

how did you get these value --- >

or, (n-1)3 = 48
or, n-1 = 16
or, n = 17
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How many integers from 0 to 50, inclusive, have a remainder  [#permalink]

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Updated on: 01 Apr 2018, 07:36
mejia401 wrote:
Calculated the same. $$\frac{Last R1 - First R1}{3} +1 = \frac{(49 - 1)}{3} +1 = 16 + 1 = 17$$.

Testing the answer choices if another multiple is needed or if there are one too many. This proved that C was the correct answer.

A. 15 * 3 = 45. 50-45 = R5. Too low.
B. 16 * 3 = 48. 50-48 = R2.
C. 17 * 3 = 51. 50-51 = R1, which is in line with what the question asks.
D. 18 * 3 = 54. 54-50= R4. Too high
E. 19 * 3 = 57. 57-50= R7.Too high

hello there

Isnt it formula of number of multiples that you used .... but if it is formula of number of multiples in X range than isnt it used when we need to find number of terms divided by 3 and not when we need to find how many integers leave remainder 1

as in my post (#7) https://gmatclub.com/forum/arithmetic-p ... l#p2035478

Originally posted by dave13 on 01 Apr 2018, 06:13.
Last edited by dave13 on 01 Apr 2018, 07:36, edited 1 time in total.
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Re: How many integers from 0 to 50, inclusive, have a remainder  [#permalink]

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01 Apr 2018, 06:25
1
dave13 wrote:
abhishekik wrote:
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

hello there

from this $$49 = 1+(n-1)3$$ I get following:

$$49 = 1+3n-3$$

$$49=3n-2$$

$$3n=51$$

how did you get these value --- >

or, (n-1)3 = 48
or, n-1 = 16
or, n = 17

Hi dave13

We have $$49 = 1+(n-1)3$$

This can be re-written as $$(n-1)3 = 49 - 1 = 48$$

Dividing by 3 on both sides, we get $$n-1 = \frac{48}{3} = 16$$

Therefore, we can arrive at n = 17

Hope this helps you!
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Re: How many integers from 0 to 50, inclusive, have a remainder  [#permalink]

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01 Apr 2018, 10:51
1
dave13 wrote:
mejia401 wrote:
Calculated the same. $$\frac{Last R1 - First R1}{3} +1 = \frac{(49 - 1)}{3} +1 = 16 + 1 = 17$$.

Testing the answer choices if another multiple is needed or if there are one too many. This proved that C was the correct answer.

A. 15 * 3 = 45. 50-45 = R5. Too low.
B. 16 * 3 = 48. 50-48 = R2.
C. 17 * 3 = 51. 50-51 = R1, which is in line with what the question asks.
D. 18 * 3 = 54. 54-50= R4. Too high
E. 19 * 3 = 57. 57-50= R7.Too high

hello there

Isnt it formula of number of multiples that you used .... but if it is formula of number of multiples in X range than isnt it used when we need to find number of terms divided by 3 and not when we need to find how many integers leave remainder 1

as in my post (#7) https://gmatclub.com/forum/arithmetic-p ... l#p2035478

Hi dave13

Yes, the formula only calculates the number of terms divided by 3. But if you observe
every(a number divisible by 3 + 1) will give a remainder of 1 when divided by 3

Here, between 0 and 50(both inclusive), there are
17 numbers which are divisible by 3 - 0(0*3) to 48(16*3)
17 numbers which leave the remainder 1(0*3 + 1) - 49(16*3 + 1)

Hope this helps clear your confusion!
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Re: How many integers from 0 to 50, inclusive, have a remainder  [#permalink]

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08 Apr 2018, 06:58
1
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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Posts: 3
Re: How many integers from 0 to 50, inclusive, have a remainder  [#permalink]

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18 Sep 2018, 10:45
50/3 = 16 (int) ----- 16*3= 48. ∴ all these 16 numbers will have a (n+1) value. E.g. 48+1= 49. And include ‘1’ = 0+1. Making the total into 17.
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Re: How many integers from 0 to 50, inclusive, have a remainder  [#permalink]

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05 Aug 2019, 00:23
From 0 to 50, both included, we have 51 numbers.
The cyclicity of 3 is 3. That means, 3 leaves only 3 reminders, 0,1,2 and the cycle repeats. You can check as below.
Reminder of 0 when divided by 3 = 0
Reminder of 1 when divided by 3 = 1
Reminder of 2 when divided by 3 = 2
Reminder of 3 when divided by 3 = 0
Reminder of 4 when divided by 3 = 1
.
.
.
So, out of 51 numbers, there are only 3 reminders possible when divided by 3. So we can divide 51 numbers into 3 sets.
1.Ones that give 0 as reminder
2.Ones that give 1 as reminder
3.Ones that give 2 as reminder

So 51/3=17.
So the final answer is C.17
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Re: How many integers from 0 to 50, inclusive, have a remainder  [#permalink]

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05 Aug 2019, 00:57
The range of numbers that leave a remainder of 1 when divided by 3:
1,4,...,49

Therefore total number of integers:
Un = U1 + (n-1)*b
49 = 1 + (n-1)*3
n=17

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

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20 Nov 2019, 18:54
Economist wrote:
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

Just list the numbers. (If you have solved enough remainder problems you know that the least possible number is remainder itself)

So list from 1,4,7,...........,40,43,47,50.

Count them.

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Re: How many integers from 0 to 50, inclusive, have a remainder   [#permalink] 20 Nov 2019, 18:54

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