Consider the following statements 1 Of two consecutive integers one is even 2 Square of an odd integ...........

Quantitative Aptitude (CDS) Number System

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Consider the following statements :

1. Of two consecutive integers, one is even.

2. Square of an odd integer is of the form 8n +1.

Which of the above statements is/are correct?

1 only

2 only

Both 1 and 2

Neither 1 nor 2

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Explanation:

Statement 1:
As we know that in two consecutive integers, the one is always odd then other is even.
Hence statement 1 is correct.
Statement 2:
By Euclid’s division
a = bq + r, 0 ≤ r < b
a and b are positive integers.
Take b = 8
Then a = 8q + r
Here, r = 0,1,2 … .7
Case (i), if r = 0
a = 8q
Then a² = 64q² = 8(8q²)
=>a² = 8 m [As m = 8q²]
Case (ii), if r = 1
a = 8q + 1 [Odd integer]
a² = (8q + 1)2
=>a² = 64q² + 16q + 1
=>a² = 8m + 1 [As, m =8q² + 2q]
Clearly, square of an odd integer is of the term 8n + 1
Hence statement 2 is correct.