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If a, b, c are consecutive positive integers and a < b < c, which of

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If a, b, c are consecutive positive integers and a < b < c, which of  [#permalink]

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New post Updated on: 18 Dec 2017, 23:06
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If a, b, c are consecutive positive integers and a < b < c, which of the following must be true?

I. c - a = 2
II. abc is an even integer
III. (a + b + c)/3 is an integer

A. I
B. II
C. I and II
D. II and III
E. I, II and III

OPEN DISCUSSION OF THIS QUESTION IS HERE: https://gmatclub.com/forum/if-a-b-and-c ... 44907.html

Originally posted by Lolaergasheva on 07 Feb 2011, 23:23.
Last edited by Bunuel on 18 Dec 2017, 23:06, edited 3 times in total.
Renamed the topic, edited the question and added the OA.
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Re: If a, b, c are consecutive positive integers and a < b < c, which of  [#permalink]

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New post 07 Feb 2011, 23:58
1
If a, b, c are consequitive positive integers and a<b<c, which of the following must be true?
1. c-a=2
2. abc is an even integer
3. (a+b+c) /3 is an integer

1.
a=a
b=a+1
c=a+2

c-a=a+2-a=2.
Correct

2. abc
a can be either even or odd
if a=even; abc=even;
if a=odd,b=even; abc=even

Rule:
multiple of any number of integers if atleast once multiplied by an even number will result in even.

even*even=even
even*odd=even

Correct

3. (a+b+c) /3 is an integer
Again;

a=a
b=a+1
c=a+2

a+b+c=(a+a+1+a+2)=3a+3

3a is always divisible by 3
3 is divisible by 3.
Thus "3a+3" is also divisible by 3.

Rule:
if integer x is divisible by n
and
integer y is divisible by n
then; x+y must be divisible by n

Correct.

Ans : "E"
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Re: If a, b, c are consecutive positive integers and a < b < c, which of  [#permalink]

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New post 08 Feb 2011, 01:22
The sum of three consecutive numbers will always be divisible by 3 <<< It's a helpful result. You might want to commit it to your memory ;)
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Re: If a, b, c are consecutive positive integers and a < b < c, which of  [#permalink]

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New post 08 Feb 2011, 03:48
Lolaergasheva wrote:
If a, b, and c are consecutive positive integers and a<b<c, which of the following must be true?
1. c-a=2
2. abc is an even integer
3. (a+b+c) /3 is an integer

A. 1
B. 2
c. 1 and 2
d. 2 and 3
e. 1, 2 and 3


1. c-a=2 --> a, b, c are consequitive positive integers and a<b<c then c=a+2 --> c-a=2. So this statement is always true;

2. abc is an even integer --> out of any 3 consecutive integers at least one must be even thus abc=even. So this statement is also always true;

3. (a+b+c)/3 is an integer --> the sum of odd number of consecutive integers is ALWAYS divisible by that odd number. So this statement is also always true. Or: (a+b+c)/3=(a+a+1+a+2)/3=(3a+3)/3=a+1=integer.

Answer: E.

AmrithS wrote:
The sum of three consecutive numbers will always be divisible by 3 <<< It's a helpful result. You might want to commit it to your memory ;)


Not only that:

• If \(k\) is odd, the sum of \(k\) consecutive integers is always divisible by \(k\). Given \(\{9,10,11\}\), we have \(k=3\) consecutive integers. The sum of 9+10+11=30, therefore, is divisible by 3.

• If \(k\) is even, the sum of \(k\) consecutive integers is never divisible by \(k\). Given \(\{9,10,11,12\}\), we have \(k=4\) consecutive integers. The sum of 9+10+11+12=42, therefore, is not divisible by 4.

• The product of \(k\) consecutive integers is always divisible by \(k!\), so by \(k\) too. Given \(k=4\) consecutive integers: \(\{3,4,5,6\}\). The product of 3*4*5*6 is 360, which is divisible by 4!=24.
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Re: If a, b, c are consecutive positive integers and a < b < c, which of  [#permalink]

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New post 18 Dec 2017, 23:07
Lolaergasheva wrote:
If a, b, c are consecutive positive integers and a < b < c, which of the following must be true?

I. c - a = 2
II. abc is an even integer
III. (a + b + c)/3 is an integer

A. I
B. II
C. I and II
D. II and III
E. I, II and III



OPEN DISCUSSION OF THIS QUESTION IS HERE: https://gmatclub.com/forum/if-a-b-and-c ... 44907.html
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Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

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Re: If a, b, c are consecutive positive integers and a < b < c, which of &nbs [#permalink] 18 Dec 2017, 23:07
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