Consider the following statements I No integer of the form 4k 3 where k is an integer can be express...........

Quantitative Aptitude (CDS) Number System

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Consider the following statements:

I. No integer of the form 4k + 3, where k is an integer, can be expressed as the sum of two squares.

II. Square of an odd integer can expressed in the form 8k + 1, where k is an integer.

Which of the above statements is/are correct?

Only I

Only II

Both I and II

Neither I nor II

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Explanation:

I. f (k) = 4k + 3
For k = 1, f (1) = 4 × 1 + 3 = 7
For k = 2, f (2) = 4 × 2 + 3 = 11
For k = 3, f (3) = 4 × 3 + 3 = 15
Values of f (k) for k = 1, 2, . . . . .. cannot be
expressed as sum of squares, since
1² + 2² = 5, 1² + 3² = 10, 2² + 3² = 13
II. f (k) = 8k + 1
For k = 1, f (1) = (8 × 1) + 1 = 9 = 3²
For k = 2, f (2) = (8 × 2) + 1 = 17
For k= 3, f (3) = (8 × 3) + 1 = 25 = 5²
For k = 4, f (4) = (8 × 4) + 1 = 33
For k = 5, f (5) = (8 × 5) + 1 = 41
f (k) = 8k + 1 is square of an odd integer only for some values of k. So, only Statement I is correct.