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?

(B) only

The correct answer is Option (3) → (B) only

To determine which statements are incorrect, let's analyze each one individually:

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

Any odd number can be represented as $2k + 1$, where $k$ is an integer. Let's square it:

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

We can factor out a $4$ from the first two terms:

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

If we let $n = k^2 + k$, the expression becomes $4n + 1$.

• Example: $3^2 = 9$. $9$ divided by $4$ leaves a remainder of $1$ ($4 \times 2 + 1$).
• Conclusion: This statement is Correct.

(B) 1 is a prime number

By mathematical definition, a prime number is a natural number greater than 1 that has exactly two distinct positive divisors: 1 and itself.

• Since $1$ only has one divisor (itself), it is not a prime number.
• It is also not a composite number. $1$ is considered a unique/unit number.
• Conclusion: This statement is Incorrect.

(C) 83356768 is divisible by 11

To check divisibility by 11, we find the difference between the sum of digits at odd positions and the sum of digits at even positions (starting from the right):

• Digits at odd positions (8, 7, 5, 3): $8 + 7 + 5 + 3 = 23$
• Digits at even positions (6, 6, 3, 8): $6 + 6 + 3 + 8 = 23$
• Difference: $23 - 23 = 0$

If the difference is $0$ or a multiple of $11$, the number is divisible by $11$.

• Conclusion: This statement is Correct.

Final Evaluation

• (A) is correct.
• (B) is incorrect.
• (C) is correct.

The correct answer is: (B) only