N is a 2-digit integer. When the digits of N are reversed

1.N=69,78
only 69 is satisfying N-m in (-1,15)
2.N=67,78,56,....
hence insufficient

Be careful, N = 69 does not satisfy the given conditions.

Cheers,
Brent

\(N = 10x + y | M = 10y + x | N - M = 9(x - y)\)
\(-1 < 9(x - y) < 15\)

1. x + y = 15 -> x = 15 - y
9(15 - y - y) is 9(15 - 2y) and is in the range, only when y = 7.
There is only one possibility for such a 2-digit number, which is 87 (Sufficient)

2. x = y + 1
There are various options for N such that this statement is true.
The numbers possible are 43,54,87 (Insufficient) (Option A)

I think we can do this without assuming any number. Let's try it.

Let's assume \(N = 10x + y\) and \(M = 10y + x\)

so \(N - M = 9(x - y)\).

Given equation becomes
\(-1 < 9(x - y) < 15\).

Think for a moment. For what value of (x-y) the above inequality would be true. Surely it should be 1 and 0.

For any other number inequality will not hold true.

So here we have \(x-y = 1\) or \(x-y = 0\)

A) The sum of N’s digits is 15.

This implies \(x + y = 15.\)

We already have one equation: \(x -y =1\).
and here statement A provides another equation. We will get to one value of N.

So it is sufficient.

We also understand that x-y =0 can't be used to get the value of x & y, because it will not give x & y as integer.

B) The tens digit of N is 1 greater its units digit.

tens digit is x and unit digit is y. So \(x-y =1.\)

Notice that this statement doesn't provide any additional detail to help us find the the number.

Hence Not Sufficient.

Ans - A.

Target question: What is the value of N?

Given: N is a 2-digit integer. When the digits of N are reversed, the resulting number is M. -1 < N – M < 15
Let x = the tens digit of N
Let y = the units digit of N
So, the VALUE of N = 10x + y

When we reverse the digits, we get M = yx
So, the VALUE of M = 10y + x

So, N - M = (10x + y) - (10y + x)
= 9x - 9y
= 9(x - y)
In other words, N - M = some multiple of 9
We're told that -1 < N – M < 15
There are exactly two multiples of 9 between -1 and 15. They are 0 and 9.
So EITHER N – M = 0 OR N – M = 9

Let's examine each case:
CASE A: If N - M = 0, then 9(x - y) = 0, which means x - y = 0, which means x = y
CASE B: If N - M = 9, then 9(x - y) = 9, which means x - y = 1, which means x = y + 1

Statement 1: The sum of N’s digits is 15
In other words, x + y = 15
If x and y are INTEGERS, and if x + y = 15, then x cannot equal y
This rules out CASE A, which means CASE B must be true. That is, x = y + 1
We now have two equations:
x + y = 15
x = y + 1

Since we COULD solve this system for x and y, we COULD determine the value of N
Since we COULD answer the target question with certainty, statement 1 is SUFFICIENT

Aside: If we solve the system, we get: x = 7 and y = 8
So, N = 78

Statement 2: The tens digit of N is 1 greater its units digit
In other words, x = y + 1
This means CASE B is true (i.e., x = y + 1 )
Given this, there are many values of N that statement 2.
For example, it could be the case that N = 21 or N = 32 or N = 43 or N = 54 etc.
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Answer: A

Cheers,
Brent

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