27^{27} - 9^{40} - 3^{79} is divisible by how many natural numbers less than 10 ?

Explanation

Simplifying the expression: 27^{27} = (3^3)^{27} = 3^{81} and 9^{40} = (3^2)^{40} = 3^{80}. The expression becomes 3^{81} - 3^{80} - 3^{79} = 3^{79}(3^2 - 3^1 - 1) = 3^{79}(9 - 3 - 1) = 3^{79} \times 5. The divisors less than 10 are 1, 3, 5, and 9. Thus, there are 4 natural numbers.

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