If z is an integer, what is the units digit of z^3?

If z is an integer, what is the units' digit of \(z^3\)?

(1) z is a multiple of 5. The units digit of a multiple of 5 can be 0 or 5, so the units digit of z^3 could be 0 or 5 as well. Not sufficient.

(2) \(\sqrt{z}\) is an integer --> z = integer^2. Clearly insufficient.

(1)+(2) The units digit of z^3 could still be 0 or 5. For example, consider z = 25 and z = 0. Not sufficient.

Answer: E.
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Solution:

Statement 1) z is a multiple of 5.

if z=5 , then units digit of z^3 is 5.

if z=10 , then units digit of z^3 is 0.

Hence Insufficient.

Statement 2) \(\sqrt{z}\) is an integer.

z is an integer then z must be perfect square.

z=4,9,25,or 36... then units digit z^3 is different for different numbers.

Hence Insufficient.

Statement 1) and Statement 2) Together.

z can be 25 or 100 which are perfect squares and are multiples of 5.

units digit of 25^3 and 100^3 is different.

Hence Insufficient.

Ans E.

Statement 1

Let us take 2 values of z which is a multiple of 5, 5 & 10. Cube of 5 will result in 5 as units digit and cube of 10 will give us a 0 as units digit.
Thus statement 1 is insufficient.

Statement 2

Let us again take 2 values, 25 & 100, which have their respective square roots as 5 & 10. This will again give us the scenario in eg. discussed in previous statement.
Thus statement 2 is also insufficient.


Combining both statements, we have again no unique values as seen in examples for statements 1 & 2.


Hence the answer is E.

Solution:

Statement 1: Z can either be 25 or 100. Insufficient.

Statement 2: Z can either be 25 or 100. Insufficient.

Combining St1 and St2,
We still have 2 choices, so its E.

St 1

Z is a multiple of 5- but what is Z? 5..10 or maybe 100? Insufficient

St 2

\(\sqrt{z}\)

This just means Z is a perfect square- obviously in suff

St 1 and St 2

Insuff- we could have 100 which is a perfect square and multiple of 5 or 25 which is a perfect square and multiple of 5

E

We need to determine the units digit of z^3.

Statement One Alone:

z is a multiple of 5.

Statement one is not enough information. If z = 5, then the units digit of z^3 = 5, and if z = 10, then the units digit of z^3 = 0.

Statement Two Alone:

z√z is an integer.

The information in statement two is not sufficient. For instance, if z = 25, then the units digit of 25^3 = 5; however, if z = 100, then the units digit of 100^3 = 0.

Statements One and Two Together:

Using the statements together, we still do not have enough information. Again, if z = 25, then the units digit of 25^3 = 5; however, if z = 100, then the units digit of 100^3 = 0.

Answer: E
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