What is the units digit of z^x, where x and z are positive integers?

IMO E
St 1) Two possibilities -
Case i) z/100 = a5bc
implies z = a5bc00;
units digit of z is zero & any power to z will have 0 as the units digit.

Case ii) z/100 = a.b5
implies z = ab5
units digit of z raised to any power will be 5

Insuff.

St 2) units digit of z^2 x z^3 = z^2 is possible when z has units digit of 1,5 & 0

Insuff.

Combining both st, we get 2 values 0 & 5.
Insuff.

Hence E.

OR if statement 1) is phrased as below, we can tick answer as A
1) z is divisible by 100 & when divided by 100 has its hundredths digit as 5
Let me know if I am missing something.

the highlighted part is not hundredths digit.

need explanation for part 2

Check the unit dights for the any positive power of 5, 6, 10,11... so statment 2 insufficient.

1. its saying that Z is divisible by 100. If any number can be completely divisible by 100 means it should have last two digits as 0. We have been asked what is the unit digit and we can clearly answer this question. Sufficient.

2. z^2 X z^3 = z^2 unit digit.
In multiplication, if we want to know unit digit number after multiplication then we have to multiply unit digits of the numbers.
for eg. 12 * 14 = unit digit will be 6
or 10 * 66 = unit digit will be 0
or 10 * 50 = unit digit will be 0
or 11 * 21 = unit digit will be 1
Last two examples are fulfilling our requirement. There are more than one unit digit i.e. 0 or 1 or maybe more. We can not reach the absolute solution. Therefore this is insufficient.

Ans is A: Only the first statement is sufficient.

Hi Ritvik,

When the first statement says that z/100 gives the hundredths digit as 5 then we have to consider only the portion after the decimal.

For e.g. 0.1234 ----> 1 is the tenths digit, 2 is the hundredths, 3 is the thousandths and 4 is the ten thousandths.

So z/100 needs to be of the form a.b5c.....

For this to happen the first number that positive integer z can take is 5, since 5/100 = 0.05, the next numbers will be 15, 25, 35, 45...., 105 and so on. Since z is a positive integer ending with 5 and x is also a positive integer, z^x will always have the units digit ending with 5. Sufficient.

Statement 2 : z^2 * z^3 has the same units digit as z^2

This implies that z^5 has the same digit as z^2. This will be possible when z has a unit digit of 1, 5, 6 and 0. Insufficient.

Hi Aditya
How did you rule out the possibiity of -
z/100 = a5bc
implies z = a5bc00;
units digit of z is zero & any power to z will have 0 as the units digit.

Option A says hundredth digit is 5 when divided by 100. Also z is a positive integer. So let's say if z = 105 , then when it is divided by 100 if will become Z = 1.05.

In Z = 1.05, 0 is tenths digit and 5 is hundredth digit.with this we confirm that z's unit digit is 5. So how much power we use on Z we get 5 as units digit everytime. So A is sufficient.

B. In B, they said units digit of Z^5 = units digit of Z^2. There are many chances for example if Z has units digit as 0, 1, 5, 6, we get same for z2 , z5 or with whatever power. So given info is not sufficient to find the units digit of Z^x

So Answer should be A

Posted from my mobile device

Interesting problem! It's already been explained, but I'll add a note about handling Statement 2 quickly.

Units digits always work in one of these ways:

- For some numbers, the units digit stays the same no matter what power you raise it to. This is true if the units digit of the original number is 0, 1, 5, or 6.
- For some numbers, the units digit alternates back and forth between two values. If you raise it to an even power, you get one units digit, and if you raise it to an odd power, you get a different units digit. This is true if the units digit of the original number is 4 or 9.
- For some numbers, the units digit changes in a cycle of 4 different values. For instance, if the original units digit is 2, then the units digits you get when you raise the number to different powers will be 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, etc. This is true if the units digit of the original number is 2, 3, 7, or 8.

The second statement is basically saying that when you raise a number to a power that's 3 higher than you originally raised it to, you get the same units digit. That won't happen if the units digit alternates back and forth between 2 options or 4 options. It'll only happen if the units digit always stays the same. So, the units digit is definitely 0, 1, 5, or 6. That isn't necessarily critical to solving this problem, but it's something that should pop into your head when you see it!