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?
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (B) only
(A) Square of an odd number is of the form 4n + 1. Correct. Any odd number is \(2k+1\), and \((2k+1)^2=4k^2+4k+1=4(k^2+k)+1\), which is one more than a multiple of 4. For example, \(3^2=9=4\times2+1\) and \(5^2=25=4\times6+1\).
(B) 1 is a prime number. Incorrect. A prime number has two different factors, 1 and itself. The number 1 has only one factor, so it is neither prime nor composite; the smallest prime is 2.
(C) 83356768 is divisible by 11. Correct. A number is divisible by 11 when the sums of its digits in alternate places differ by 0 or a multiple of 11. Counting places from the right:
Digits in the odd places (8, 7, 5, 3): \(8+7+5+3=23\)
Digits in the even places (6, 6, 3, 8): \(6+6+3+8=23\)
Difference \(=23-23=0\), so the number is divisible by 11