The probability that a randomly selected 2-digit number belongs to the set n ∈ N : (2ⁿ - 2) is a multiple of 3 is equal to :

Solution

Maths sir · 4 steps

Write 2 as 3 minus 1, and the whole condition collapses to one thing: n must be odd.

1. Work modulo 3. Since 2 ≡ -1 (mod 3), every power of 2 becomes a power of -1. That gives the remainder at once.

   2ⁿ - 2 ≡ (-1)ⁿ + 1 (mod 3)

2. Check odd and even. For odd n, (-1)ⁿ + 1 = 0, so 2ⁿ - 2 is a multiple of 3. For even n, it is 2, so it is not. For example, n = 3 gives 6, while n = 4 gives 14.
3. Count favourable cases. The two-digit numbers are 10 to 99, which is 90 numbers. Exactly half of them, 45, are odd.
4. So. The probability is 45/90 = 1/2, which is option (c).

The idea to keep. For divisibility of aⁿ by m, replace a by a small remainder such as -1 or 1. The pattern repeats, and counting becomes easy.

Answer from the NTA final answer key. Solution by PrepWiser.