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 to calculate the unit digits please help me.
greenlight-admin's picture

Just use a calculator... It goes very fast and you're entitled to one in the exam
greenlight-admin's picture

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.

For more information about the GRE's onscreen calculator watch this video: https://www.greenlighttestprep.com/module/general-gre-info-and-strategie...

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.
greenlight-admin's picture

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

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
greenlight-admin's picture

That is, indeed, the approach.

Cheers,
Brent

What do you guys mean 35/4= 8 remainder 3? it's 8.75. Where does the 3 come in?
greenlight-admin's picture

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 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
greenlight-admin's picture

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
What is the unit digit of 2^96
greenlight-admin's picture

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) answer. I think it's all about estimation.
thanks
greenlight-admin's picture

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?

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