Unit Digit

GRE Quant Problem Solving
1

“I’m Overwhelmed” Tickbox Quix #2 - Question 15:

What is the remainder of 3^40 divided by 5?

I understand that I have to look for the pattern of the following:
3^1 = 3
3^2 = 9
3^3 = 27
3^4 = 81

Greg states that since 40 is a multiple of 4, we know the unit digit will be 1. But 40 is also a multiple of 2, so how can we rule out 9 as being the unit digit?

Thanks in advance for the help.

2

Hi, so in such questions we try to find the cyclicity i.e. after how many times will the pattern repeat.

So for 3, the cyclicity is 4,
3^1 = 3
3^2 = 9
3^3 = 27
3^4 = 81

3^5 = 243
3^6 = 729
3^7 = 2187
3^8 = 6561

So the units digit pattern is (3,9, 7, 1) and this repeats again.

  1. If we get a power of the form 4n + 1, unit’s digit is 3
  2. If we get a power of the form 4n + 2, unit’s digit is 9
  3. If we get a power of the form 4n + 3, unit’s digit is 7
  4. If we get a power of the form 4n + 4, unit’s digit is 1 (40 comes in this case for n=9)

We do not consider 2 as a potential divisor because this pattern repeats after every 4th number not after every 2nd number. Here, for example, we can say that the unit’s digit for 3^1 and 3^5 is same but not for 3^1 and 3^3