Consider the number 2ⁿ, where n is a natural number. Check whether there is any value of n for which 2ⁿ is divisible by 3?

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Solution

Step1:
Prime Factorization of 2: $2 = 1 \times 2$
Prime Factorization of 2ⁿ : $2^{n} = (1^{n} \times 2^{n}) = 1^{n} \times 2^{n}$
Thus, the prime factorization of 2ⁿ does not contain 3.
Hence, there is no value of n for which 2ⁿ is divisible by 3.

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Consider the number 2n, where n is a natural number. Check whether there is any value of n for which 2n is divisible by 3?

Step1:Prime Factorization of 2: 2=1×2Prime Factorization of 2n : 2n=1n×2n=1n×2nThus, the prime factorization of 2n does not contain 3.Hence, there is no value of n for which 2n is divisible by 3.

Consider the number 2ⁿ, where n is a natural number. Check whether there is any value of n for which 2ⁿ is divisible by 3?

Step1:
Prime Factorization of 2: $2 = 1 \times 2$
Prime Factorization of 2ⁿ : $2^{n} = (1^{n} \times 2^{n}) = 1^{n} \times 2^{n}$
Thus, the prime factorization of 2ⁿ does not contain 3.
Hence, there is no value of n for which 2ⁿ is divisible by 3.