Consider the following statements

(A) Square of an odd number is of the form 4n + 1.

(B) 1 is a prime number.

(C) 83356768 is divisible by 11.

Which of the statement(s) given above is/are incorrect?

Statement A: Square of an odd number is of the form 4n + 1

Testing with examples:

• $1^2 = 1 = 4(0) + 1$
• $3^2 = 9 = 4(2) + 1$
• $5^2 = 25 = 4(6) + 1$
• $7^2 = 49 = 4(12) + 1$

Any odd number can be written as $2k + 1$ where $k = 0, 1, 2, 3...$

$(2k + 1)^2$

$= 4k^2 + 4k + 1$

$= 4(k^2 + k) + 1$

This is in the form $4n + 1$ where $n = k^2 + k$.

Statement A is correct.

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Statement B: 1 is a prime number

A prime number must satisfy two conditions:

1. It should be a natural number greater than 1
2. It should have exactly two divisors: 1 and itself

For the number 1:

• Divisors of 1 = only {1}
• 1 has only one divisor, not two
• Also, 1 is not greater than 1

By definition, 1 is not a prime number.

Statement B is incorrect.

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Statement C: 83356768 is divisible by 11

Divisibility rule for 11: Add digits at odd positions (from left), add digits at even positions (from left). If their difference equals 0 or is divisible by 11, then the number is divisible by 11.

For 83356768:

Position 1, 2, 3, 4, 5, 6, 7, 8

Digit: 8, 3, 3, 5, 6, 7, 6, 8

Odd positions (1, 3, 5, 7): $8 + 3 + 6 + 6 = 23$

Even positions (2, 4, 6, 8): $3 + 5 + 7 + 8 = 23$

Difference: $23 - 23 = 0$

Since the difference equals 0, the number is divisible by 11.

Statement C is correct.

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Only statement (B) is incorrect because 1 is not a prime number.

Answer: (B) only