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# If the sum of x distinct, positive integers is less than 75, what is

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Math Expert
Joined: 02 Sep 2009
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If the sum of x distinct, positive integers is less than 75, what is  [#permalink]

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19 Feb 2019, 21:57
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If the sum of x distinct, positive integers is less than 75, what is the greatest possible value of x ?

A. 8
B. 9
C. 10
D. 11
E. 12

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If the sum of x distinct, positive integers is less than 75, what is  [#permalink]

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Updated on: 21 Feb 2019, 07:32
Bunuel wrote:
If the sum of x distinct, positive integers is less than 75, what is the greatest possible value of x ?

A. 8
B. 9
C. 10
D. 11
E. 12

sum of distinct integers < 75
1+2+3... +11;
IMO D ;11 will be the largest integer..
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Originally posted by Archit3110 on 20 Feb 2019, 00:48.
Last edited by Archit3110 on 21 Feb 2019, 07:32, edited 1 time in total.
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Re: If the sum of x distinct, positive integers is less than 75, what is  [#permalink]

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21 Feb 2019, 07:30
Archit3110 wrote:
Bunuel wrote:
If the sum of x distinct, positive integers is less than 75, what is the greatest possible value of x ?

A. 8
B. 9
C. 10
D. 11
E. 12

IMO e
12
3+4+5+....+12= 75

Bunuel ; is the given answer option correct? if so please provide solution..

First of all notice that we are told that the sum is LESS than 75. Next, the question asks about the greatest possible NUMBER of distinct, positive integers the sum of which is LESS than 75.

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 = 66 < 75. If you add the next one, 12, then the sum would be 78. So, the greatest possible NUMBER of distinct, positive integers the sum of which is LESS than 75 is 11 (from 1 to 11, inclusive).

Hope it helps.
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Re: If the sum of x distinct, positive integers is less than 75, what is  [#permalink]

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21 Feb 2019, 08:05
Bunuel wrote:
If the sum of x distinct, positive integers is less than 75, what is the greatest possible value of x ?

A. 8
B. 9
C. 10
D. 11
E. 12

$$\frac{n(n+1)}{2} < 75$$

Now check using the given options...

$$11*\frac{12}{2} = 66 < 75$$ , Answer ,must be (D)
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Re: If the sum of x distinct, positive integers is less than 75, what is  [#permalink]

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21 Feb 2019, 09:21
Bunuel wrote:
If the sum of x distinct, positive integers is less than 75, what is the greatest possible value of x ?

A. 8
B. 9
C. 10
D. 11
E. 12

we are looking for the number of integers whose sum is less then 75.

Number of integers = x. we are asked to maximize x.

All the integers are positive. Thus we must start with 1 and continue the series with lower integers to get the maximum value of x.

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 = 66 which is less than 75. It is not possible to add any other integers.

Thus, x =11.

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If the sum of x distinct, positive integers is less than 75, what is  [#permalink]

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22 Feb 2019, 09:52
What if the numbers are 2,4,5,6,7,8,9,10,11,12? The sum of these numbers is 74. The question hasn't said the numbers are consecutive? It just said x distinct positive integers, that's it! Why is this case not possible? Why can't E be the choice?
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Re: If the sum of x distinct, positive integers is less than 75, what is  [#permalink]

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22 Feb 2019, 09:53
Quote:
If the sum of x distinct, positive integers is less than 75, what is the greatest possible value of x ?

A. 8
B. 9
C. 10
D. 11
E. 12

What if the numbers are 2,4,5,6,7,8,9,10,11,12? The sum of these numbers is 74. The question hasn't said the numbers are consecutive? It just said x distinct positive integers, that's it! Why is this case not possible? Why can't E be the choice?
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Chaitanya

+1 Kudos

if you like my explanation!!!

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Re: If the sum of x distinct, positive integers is less than 75, what is  [#permalink]

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22 Feb 2019, 10:07
1
eswarchethu135 wrote:
Quote:
If the sum of x distinct, positive integers is less than 75, what is the greatest possible value of x ?

A. 8
B. 9
C. 10
D. 11
E. 12

What if the numbers are 2,4,5,6,7,8,9,10,11,12? The sum of these numbers is 74. The question hasn't said the numbers are consecutive? It just said x distinct positive integers, that's it! Why is this case not possible? Why can't E be the choice?

Because in your explanation the value of x is 10. This is because the question is asking for the maximum numbers of distinct integers that can be added to get a sum<75. The question is not asking for the greatest of those integers.
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Re: If the sum of x distinct, positive integers is less than 75, what is  [#permalink]

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22 Feb 2019, 10:10
Shivam710 wrote:
eswarchethu135 wrote:
Quote:
If the sum of x distinct, positive integers is less than 75, what is the greatest possible value of x ?

A. 8
B. 9
C. 10
D. 11
E. 12

What if the numbers are 2,4,5,6,7,8,9,10,11,12? The sum of these numbers is 74. The question hasn't said the numbers are consecutive? It just said x distinct positive integers, that's it! Why is this case not possible? Why can't E be the choice?

Because in your explanation the value of x is 10. This is because the question is asking for the maximum numbers of distinct integers that can be added to get a sum<75. The question is not asking for the greatest of those integers.

Got it. Thanks for clarification.

Posted from my mobile device
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Re: If the sum of x distinct, positive integers is less than 75, what is  [#permalink]

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24 Feb 2019, 06:46
Bunuel wrote:
If the sum of x distinct, positive integers is less than 75, what is the greatest possible value of x ?

A. 8
B. 9
C. 10
D. 11
E. 12

Let’s analyze the answer choices in reverse order since we are looking for the greatest possible value of x.

E. 12

Since we want x to be as large as possible, we want the smallest of the x integers to be 1. Therefore, if x = 12, the sum of the first 12 positive integers is:

1 + 2 + 3 + … + 12 = 12 x (1 + 12)/2 = 6 x 13 = 78

This is not less than 75, so x can’t be 12.

D. 11

We see that 1 + 2 + 3 + … + 11 = 78 - 12 = 66.

This is less than 75, thus the greatest possible value of x is 11.

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Re: If the sum of x distinct, positive integers is less than 75, what is   [#permalink] 24 Feb 2019, 06:46
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