Consider the following statements:
I. The unit digit of \(7^{24}\) is 1.
II. When \(2^{10}\) is divided by 7, the remainder is 2.
III. The number 36 has exactly 8 factors.
Which of the above statements is/are correct?
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → I and II only
Unit digits of powers repeat in a cycle; a remainder can be read from the actual value; the number of factors of \(p^a\times q^b\) is \((a+1)(b+1)\).
I. The unit digit of \(7^{24}\) is 1. — Correct. The unit digits of powers of 7 repeat every 4 powers.
Unit digits of \(7^1,7^2,7^3,7^4\) are 7, 9, 3, 1
24 is a multiple of 4, so \(7^{24}\) ends like \(7^4\), in 1
II. When \(2^{10}\) is divided by 7, the remainder is 2. — Correct. 1024 is 2 more than a multiple of 7.
\(2^{10}=1024\)
\(7\times146=1022\)
Remainder \(=1024-1022=2\)
III. The number 36 has exactly 8 factors. — Incorrect. 36 has 9 factors; as a perfect square it has an odd number of factors.
\(36=2^2\times3^2\)
Number of factors \(=(2+1)(2+1)=9\)
They are 1, 2, 3, 4, 6, 9, 12, 18, 36