I don't understand this

Question

no whole number power of three has units digit

a)1

b)3

c)6

d)9

Answer 1

I realized after typing this that that the question I answered was "no whole number TO the power of three has a _ in the units digit". Whoops. Yeah, the other guys' answer is way better, LOL. But this is neat too if you're interested. 

 

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A whole number to the power of 3 looks like 10³. This means 10x10x10. 

 

They are asking you to make an observation about what happens to whole numbers when you multiply them by themselves three times.

 

1x1x1 = 1

 

2x2x2 = 8

 

3x3x3 = 27

 

4x4x4 = 64

 

5x5x5 = 125

 

...

 

and so on.

 

The units digit of 1 is 1. 

 

The units digit of 8 is 8.

 

The units digit of 27 is 7. 

 

The units digit of 64 is 4.

 

The units digit of 125 is 5.

 

(By units digit they mean the number in the "one's place".)

 

If you've taken algebra, we can think of what happens to a number written in the form a + b when you cube it (raise it to the third power, multiply it by itself three times, all the same thing).

 

(a + b)³ = a³ + 3a²b + 3ab² + b³.

 

In our number system, we group things in tens. We have a ones place and a tens place and a hundreds place and so on.

 

Note that 137 can be written as 100 + 30 + 7 or 10² + 3(10) + 7.

 

Let's write it as 130 + 7. That's probably the most helpful. So we have 13(10) + 7 and when we cube this number, referring to the binomial we cubed above, it will look like some multiple of 10 plus 7³.

 

That is, we only need to consider the 7 in 137 to know what the units digit of 137³ is. If we take 7³ we get 343. So we would be adding that 343 onto some multiple of ten. Since 343 itself is 34(10) + 3, this is the same as adding the 3 onto a multiple of ten. SO that 3 IS the units digit of 137³.

 

More Generally: Any number in our base 10 number system can be written in the form A(10) + B. So when we cube this we get that (A(10) + B)³ = something times 10 + B³. Then the units digit of B³ is the units digit of that whole number cubed. 

 

For any number we take, the only units digits possible in our number system are the integers 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. 

 

In Short: To answer this question, see what you get when you cube the numbers 0 through 9. The numbers you get in the ones place when you do are the only ones that are possible to get in the ones place when you cube ANY number.

 

6x6x6 = ?

 

7x7x7 = ?

 

8x8x8 = ?

 

9x9x9 = ?

 

0x0x0 = ?

 

 

From the ones I did in the beginning, so far you know that you can get a 1, an 8, a 7, a 4, and a 5. See what pops up as the right-most digit when you multiply out the numbers above. Any that don't appear that are listed in the question will be your answer. 

 

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Sorry if there are any typos or mistakes.

Answer 2

Hey Kaylie,

 

It looks like you have to identify how each number will be expressed:

 

1 = 3^0

3 = 3^1

9 = 3^2

 

If you express 6 = 3^N, then N will be between 1 and 2 (as decimal or fractional) (which means not a whole number).

 

It looks like, the answer will be c.