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## Comment on

Consecutive Integers## Hello

In below question

N = abc where a, b and c are the hundreds, tens and units digit respectively. If a, b and c are non-zero consecutive numbers such that a < b < c, then which of the following must be true?

I.N is always divisible by 2

II.N is always divisible by 3

III.N is divisible by 6 only if b is odd.

If we apply the logic (n-1)n(n+1) then it does not hold true for above case.

Can you help me understand why not applicable to above case.

Regards,

Abhimanyu

## Be careful. abc does not

Be careful. abc does not represent the product of three numbers.

Each variable (a, b, and c) represents a DIGIT in the 3-digit number abc. As such the rules related to the product (n-1)n(n+1) do not hold.

## Then how it can be solved

## The question: N = abc where a

The question: N = abc where a, b and c are the hundreds, tens and units digit respectively. If a, b and c are non-zero consecutive numbers such that a < b < c, then which of the following must be true?

I. N is always divisible by 2

II. N is always divisible by 3

III. N is divisible by 6 only if b is odd.

------------------------------------------

IMPORTANT:

Property #1) If an integer is divisible by 3, then the sum of its integers will be divisible by 3.

Property #2) If an integer is divisible by 6, then it must be divisible by 3 AND 2.

To get a feel for the properties of these 3 consecutive integers, let's...

Let x = a the 1st number (aka a)

So, x+1 = the 2nd number (aka b)

And x+2 = the 3rd number (aka c)

So, the sum a + b + c = x + (x+1) + (x+2) = 3x + 3 = 3(x + 1)

Aha, so the SUM of any 3 consecutive integers must be divisible by 3 (since 3(x+1) is definitely divisible by 3)

Now let's examine the statements...

I. N is always divisible by 2

This need not be true

For example, one possible value of N is 345, in which case N is NOT divisible by 2

II. N is always divisible by 3

This IS true

We already showed that, since the SUM of any 3 consecutive integers must be divisible by 3, then we know that N is definitely divisible by 3

III. N is divisible by 6 only if b is odd.

If b is odd, then c is EVEN, which means N is divisible by 2

Since N is also divisible by 3, we can conclude that N IS divisible by 6

So statements II and III must be true.

## Will the GMAT always

## Yes, on test day, the

Yes, on test day, the question will explicitly state the kinds of consecutive numbers.

Consecutive integers ...-3, -2, -1, 0, 1, 2, 3, ...

Consecutive EVEN integers: ...-6, -4, -2, 0, 2, 4, 6, ...

Consecutive ODD integers: ...-5, -3, -1, 1, 3, 5,...

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