# Question: Units Digit of 53 to Power of 35

## Comment on Units Digit of 53 to Power of 35

### Please I don't understand how ### Just use a calculator... It

Just use a calculator... It goes very fast and you're entitled to one in the exam ### That's a good idea, but it

That's a good idea, but it won't work. First of all, the onscreen calculator is very basic, and does not have an exponent button. This means you must enter 53 x 53 x 53 x 53 x 53 x 53 x 53 x .... (35 times). If you miscount, your answer will be incorrect.

So that's one problem. An even bigger problem with using the onscreen calculator is that it cannot display more than 8 digits. If a number has more than 8 digits, the calculator displays an error message. This is a problem because 53^35 is a 61-digit number.

### I meant use the calculator to

I meant use the calculator to do the calculations up until we find the repeating sequence, for example, here we would only need to do 53x53x53x53 which I hope can be handled by the GRE calculator. ### Ah, sorry for the

Ah, sorry for the misunderstanding. Yes, you can certainly use the calculator to find the pattern.

### The easiest approaach will be

The easiest approaach will be to first determine the number of cycle(c), and then divide exponent(n) by the number of cycle.In this case ( n/c ; 35/4 = 8 , remainder 3 ). Now since you have a remainder of three. Your answer is 7 because it is the third number of the cycle you have determined ### That is, indeed, the approach

That is, indeed, the approach.

Cheers,
Brent

### What do you guys mean 35/4= 8

What do you guys mean 35/4= 8 remainder 3? it's 8.75. Where does the 3 come in? ### Here's my video introduction

Here's my video introduction to remainders: https://www.greenlighttestprep.com/module/gre-integer-properties/video/840

It's true to say that 35/4 = 8.75.
So, for example, if 4 people were to share in the cost of a \$35 gift, then the amount each person pays = \$8.75
Of course, this works with money, since we can easily divide one dollar into smaller amounts.

However, what if we were to evenly distribute 35 puppies among 4 children?
In this case, we can't say that each child gets 8.75 puppies (at least I hope not!!!).
In this case, we could give each child 7 puppies, and have 3 puppies remaining.

Cheers,
Brent

### If I have 19 to the power of

If I have 19 to the power of 42
it has a cycle of 2, so 42/2= 21 so the unit digit will be 1, the last digit of the result?

if it will be the case of 2 to the power of 43
it has a cycle of 4, 43/4= 10.75 so the remainder will be the unit number = 3.

3 to the power of 44
it has a cycle of 4, 44/4=11 so the unit number is 1.

but in the case of 52 to the power of 12 with a cycle of 4, 12/4=3... and the unit number is 6, so this approach in which cases may apply?

Thank you ### Sorry, but that approach won

Sorry, but that approach won't work for most/many cases.

Let's find the units digit of 19^42 first.
19^1 = --9
19^2 = --1
19^3 = --9
19^4 = --1
19^5 = --9
19^6 = --1
This has cycle 2, so when the exponent is a multiple of 2, the units digit is 1.
Since 42 is a multiple of 2, the units digit of 19^42 is 1
-----------------------------------------

Let's find the units digit of 2^43 next.
2^1 = ---2
2^2 = ---4
2^3 = ---8
2^4 = ---6
2^5 = ---2
2^6 = ---4
2^7 = ---8
2^8 = ---6
2^9 = ---2

This has cycle 4, so when the exponent is a multiple of 4, the units digit is 6.
Since 40 is a multiple of 4, the units digit of 2^40 is 6
Now let's continue the 6-2-4-8-6-2-4-8 pattern
The units digit of 2^41 is 2
The units digit of 2^42 is 4
The units digit of 2^43 is 8
---------------------------------------------

Let's find the units digit of 3^44 next.
3^1 = ---3
3^2 = ---9
3^3 = ---7
3^4 = ---1
3^5 = ---3
3^6 = ---9
3^7 = ---7
3^8 = ---1
3^9 = ---3

This has cycle 4, so when the exponent is a multiple of 4, the units digit is 1.
Since 44 is a multiple of 4, the units digit of 3^44 is 1
------------------------------------------

Here's an article with extra practice at the end: https://www.greenlighttestprep.com/articles/units-digits-big-powers

Cheers,
Brent

### Dear hi

Dear hi
What is the unit digit of 2^96 ### Let's start by listing some

Let's start by listing some powers of 2:
2^1 = ---2
2^2 = ---4
2^3 = ---8
2^4 = ---6
2^5 = ---2
2^6 = ---4
2^7 = ---8
2^8 = ---6
2^9 = ---2
...etc

Notice that the cycle = 4

Also notice that, when the exponent is a multiple of 4, the units digit is 6.
Some examples: 2^4 = --6, 2^8 = --6, 2^12 = --6, 2^16 = --6, etc

Since 96 is a multiple of 4, we know that 2^96 = --6
So, the units digit is 6

Cheers,
Brent

### Did I get my previous ( 53^35

Did I get my previous ( 53^35) answer. I think it's all about estimation.
thanks ### Yes, I saw your comment at

Yes, I saw your comment at https://www.greenlighttestprep.com/module/gre-powers-and-roots/video/1034
However, 53 has cycle 4 (not 5), as I showed in that post.

I'm just not sure what you mean by estimation. We certainly don't want to estimate the value of 53^35
We need to make some key observations:

1) 53 has a cycle of 4 (that is it repeats itself every four powers)
2) The unit's digit of 53^n is 1 whenever n is a multiple of 4 (4 being the cycle of powers of 53)
For example, the unit's digit of 53^4 is 1, the unit's digit of 53^8 is 1, the unit's digit of 53^12 is 1, the unit's digit of 53^16 is 1, etc

Since 32 is a multiple of 4, we know that the unit's digit of 53^32 is 1

Continuing our pattern we get:
The unit's digit of 53^33 is 3
The unit's digit of 53^34 is 9
The unit's digit of 53^35 is 7

Does that help?