Patterns, Relations, and Function

[Momspn]

I am trying to help my son with his pre-algebra, and of course all I have is a workbook not the actual math book.

The problem is:

A computer multiplied 100 nines. You can use patterns to find the ones digit of the product.
Find the ones digit for the product of : 1 nine (answer 81) 2 nines (answer 729) 3 nines (answer 6561) 4 nines (59049)
Describe the pattern. (answer- multiply by 9)
What is the ones digit of the computer's product? (answer-81)
Use the above method to find the ones digit of the product when 4 is multiplied by itsel 100 times. (answer- 40,000,000)

I have no clue if his answers are correct. Can anyone help me??

 

[soroban]

Hello, Momspn!

You are both misreading the problem . . . All answers are wrong, sorry.

A computer multiplied 100 nines. You can use patterns to find the ones-digit of the product.
The ones-digit is the last (rightmost) digit.

Find the ones-digit for the product of: 1 nine, 2 nines, 3 nines, 4 nines.
Describe the pattern.

\(\displaystyle \begin{array}{cccccc} 9^n & \text{product} & \text{ones-digit} \\ \hline 9 & 9 & 9 \\ 9^2 & 81 & 1 \\ 9^3 & 729 & 9 \\ 9^4 & 6561 & 1 \\ \vdots & \vdots & \vdots \end{array}\)

The ones-digits alternate between 1 and 9.
. . If n is odd, the ones-digit is 9.
. . If n is even, the ones-digit is 1.

What is the ones-digit of the computer's product?

\(\displaystyle \text{We have: }\:9^{100}\)

\(\displaystyle \text{Since }n=100\text{ is }even\text{, the ones-digit is }1.\)

Use the above method to find the ones-digit of the product when 4 is multiplied by itself 100 times.

\(\displaystyle \begin{array}{cccccc} 4^n & \text{product} & \text{ones-digit} \\ \hline 4& 4 & 4 \\ 4^2 & 16 & 6 \\ 3^3 & 64 & 4 \\ 4^4 & 256 & 6 \\ \vdots & \vdots & \vdots \end{array}\)

The ones-digits alternate between 4 and 6.
. . If n is odd, the ones-digit is 4.
. . If n is even, the ones-digit is 6.

\(\displaystyle \text{We have: }\:4^{100}\)

\(\displaystyle \text{Since }n = 100\text{ is }even\text{, the ones-digit is 6.}\)

 

[mmm4444bot]

Hi Mom:

It looks to me like your son does not understand the exercise.

Let's start with a couple of definitions.

The "ones digit" is the rightmost digit, in any whole number.

EG:

23987 --> the "ones digit" is 7.

238 --> the "ones digit" is 8.

2390665 --> the "ones digit" is 5.

21340982350928644 --> the "ones digit" is 4.

6 --> the "ones digit" is 6.

Next, let's understand what's meant by multiplying 100 nines. It means that there are 100 factors, and each of these factors is 9.

EG:

9 × 9 = 81 --> This is the product of two nines (two factors of 9).

9 × 9 × 9 = 729 --> This is the product of three nines (three factors of 9).

9 × 9 × 9 × 9 = 6561 --> This is the product of four nines (four factors of 9).

It's a common mistake among educators to state that 81 is what we get by "multiplying nine by itself twice". This is false. Nine is only being multiplied by itself ONCE.

Likewise, it's wrong to say that 729 is what we get by "multiplying nine by itself three times". This is false. Nine is only being multiplied by itself TWICE.

I think it's much clearer to discuss these products in terms of how many factors there are, instead of saying how many times the first factor is multiplied by itself.

For example, with two factors of nine, the product is 81.

With three factors of nine, the product is 729.

With one factor of nine, we simply have 9.

To find the pattern, your son should write down and study a table, such as the following. What conclusion can be made regarding the relationship between the number of factors of nine and the ones digit of the product?

1 factor: 9
2 factors: 81
3 factors: 729
4 factors: 6561
5 factors: 59049
6 factors: 531441
7 factors: 4782969
8 factors: 43046721
9 factors: 387420489
10 factors: 3486784401
11 factors: 31381059609
12 factors: 282429536481

Hint: Think in terms of even numbers of factors versus odd numbers of factors.

This should be enough to deduce the ones digits of the product of 100 factors of nine. (By the way, this product is 61 digits long!)

Use the same strategy for finding a pattern in the ones digit of products involving factors of 4.

Again, it's not technically correct when your son's teacher says "4 is multiplied by itself 100 times". What the teacher is trying to discuss is the product that comes from multiplying 100 factors of 4 together (i.e., 4 is multiplied by itself 99 times).

If I wrote anything that you do not understand, then please ask specific questions. We're happy to provide more explanations and examples.

 

[Momspn]

Thank you both for all the help. I think he understands it better the way you explained it. I am trying to learn at the same time. So, again, Thank you.