anceer wrote:

If a is an integer, what is the units digit of a^18?

1. a^2 has a units digit of 9

2. a^7 has a units digit of 3

The cyclicity of the last digit is at most 4.

e.g.

Last digit 2 - 2, 4, 8, 6 ... (cycle of 4)

Last digit 3 - 3, 9, 7, 1 ... (cycle of 4)

Last digit 4 - 4, 6, 4, 6 ... (cycle of 2)

Last digit 5 - 5, 5, 5, 5 ... (cycle of 1)

Last digit 6 - 6, 6, 6, 6 ... (cycle of 1)

Last digit 7 - 7, 9, 3, 1 ... (cycle of 4)

Last digit 8 - 8, 4, 2, 6 ... (cycle of 4)

Last digit 9 - 9, 1, 9, 1 ... (cycle of 2)

If last digit is 0 or 1, it remains 0 or 1 - cycle of 1

1. \(a^2\) has a units digit of 9

In the cycle of 4, \(a^2\) represents the second term. Last digit of a square is 9 in two cases.

Last digit 3 - 3, 9, 7, 1

Last digit 7 - 7, 9, 3, 1

So we don't know the last digit of a. But we are not asked the last digit of a. We are asked the last digit of \(a^{18}\). It is also the second term in the cycle of 4. So it doesn't matter whether last digit of a is 3 or 7, the last digit of \(a^{18}\) will be 9 in either case. Sufficient alone.

2. \(a^7\) has a units digit of 3

In the cycle of 4, \(a^7\) represents the third term. Last digit of the third term will be 3 in only one case - when the last digit of a is 7. So we can say that the last digit of \(a^{18}\) must be 9. Sufficient alone.

Answer (D)

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