Is the positive integer x an even number? (1) If x is divided by 5

Ans: E
1)x=5a+2
a=1;x=7 not even : a=2; x=12 x=even not sufficient

2)x=7b+1
b=1; x=8 x=even: b=2;x=15 x=odd not sufficient

combining=>x=35q+2....x can be 37,72
so both even and odd are possible
Not sufficient

Step 1: Analyse Question Stem

It is known that x is a positive integer.
We have to find out if it is even.
An even number can be identified by its units digit, which will always be 2, 4, 6, 8 or 0.

Step 2: Analyse Statements Independently (And eliminate options) – AD / BCE

Statement 1: If x is divided by 5, the remainder is 2.

Using the division algorithm, x can be written as, x = 5p + 2, where p is a non-negative integer.

This means that x can be 2 or 7 or 12 or 17 and so on. In some cases, x is even and in some cases, it is odd.

The data in statement 1 is insufficient to find out if x is even.
Statement 1 alone is insufficient. Answer options A and D can be eliminated.

Statement 2: If x is divided by 7, the remainder is 2.

Using the division algorithm, x can be written as, x = 7q + 2, where q is a non-negative integer.

This means that x can be 2 or 9 or 16 or 23 and so on. In some cases, x is even and in some cases, it is odd.

The data in statement 2 is insufficient to find out if x is even.
Statement 2 alone is insufficient. Answer option B can be eliminated.

Step 3: Analyse Statements by combining

From statement 1: x = 5p + 2

From statement 2: x = 7q + 2

Therefore, 5p = 7q or \(\frac{ p }{ q}\) = \(\frac{7 }{ 5}\). This means that p and q are multiples of 7 and 5 respectively.

If p = 7, x = 5 * 7 + 2 = 37; here, x is odd.

If p = 14, x = 5 * 14 + 2 = 72; here x is even.

The combination of statements is insufficient to answer if x is even.
Statements 1 and 2 together are insufficient. Answer option C can be eliminated.

The correct answer option is E.