unit digit

[vipulgoyal]

what id the unit digit of P^xy

1. unit digit of p^x = 1
2. unit digit of p^y = 1

[Anju@Gurome]

what is the units digit of p^(xy)

1. Units digit of p^x = 1
2. Units digit of p^y = 1

The tricks to solve this problem are...

• #1. Units digit of any integral powers of any integer with units digit 1 is always 1.
  #2. p^(xy) = (p^x_^y = (p^y)^x

Statement 1: p^(xy) = (p^x)^y
As units digit of p^x is 1, units digit of (p^x)^y is also 1.

Sufficient

Statement 2: p^(xy) = (p^y)^x
As units digit of p^y is 1, units digit of (p^y)^x is also 1.

Sufficient

The correct answer is D.

[Matt@VeritasPrep]

Just to add a note of caution, the answer actually isn't D, at least not as worded: there's nothing in the stem implying that x and y are integers.

Take statement (1) alone, for instance. If p = 3 and x = 4, then p^x = 81, which has a units digit of 1. But if y is equal to 1/4, then p^(xy) = 3^(4 * 1/4) = 3^1 = 3.

[spoiler]I think the answer is E: one counterexample showing that the units digit of p^(xy) need not be 1 is p = 121, x = 1/2, y = 1/2.[/spoiler]

That said, a lot of definitions of "units digit" seem to restrict this definition to integers: "the units digit is the rightmost digit of the integer". The GMAT may follow this custom (I've never seen them ask about the "units digit" of a non-integer), which would force p^(xy) to be an integer, invalidating my counterexample (though the answer still isn't D.)

[MBAsa]

Thanks for that Matt. Totally missed the interger bit

[Blue_Skies]

Thanks Matt.This is another reminder of not to consider any number integer unless otherwise stated.