Number Theory type of problem--need help understanding number relationships. yikes.

• Dec 18th 2008, 04:41 PM
eri
Number Theory type of problem--need help understanding number relationships. yikes.
They say, 100 is a multiple of 4, so that's why the units digit of 3100 is the same as that of 34 However, 100 is also a multiple of 5. So why did they pick 4?

If n is an integer with units digit 3,
what is the units digit of n 100 ?

Since only the units digit of n matters, you can start by exploring what happens when n equals 3. Raise 3 to powers and observe the units digits of the results.
31 = 3
32 = 9
33 = 27
34 = 81
35 = 243

units digit:
3
9
7
1
3

Notice that the units digit of 35 is again 3, so from here on the sequence will repeat itself and continue to cycle through the four values 3, 9, 7, 1. Since 100 is a multiple of 4, the units digit of 3100 is the same as that of 34
• Dec 18th 2008, 05:43 PM
Soroban
Hello, eri!

You should indicate your exponents more carefully.

Quote:

They say, 100 is a multiple of 4, that's why the units digit of $3^{100}$ is the same as that of $3^4$.
However, 100 is also a multiple of 5. So why did they pick 4?

If $n$ is an integer with units digit 3, what is the units digit of $n^{100}$ ?

Since only the units digit of $n$ matters, you can start by exploring what happens when $n = 3.$
Raise 3 to powers and observe the units digits of the results.

. . $\begin{array}{cccc}& & & \text{units digit} \\ \hline
3^1 &=& 3 & 3\\ 3^2 &=& 9 & 9\\ 3^3 &=& 27 & 7\\ 3^4 &=& 81 & 1\\ 3^5 &=& 243 & 3 \end{array}$

Notice that the units digit of $3^5$ is again 3, so from here on the sequence
will repeat itself and continue to cycle through the four values 3, 9, 7, 1.

Since 100 is a multiple of 4, the units digit of $3^{100}$ is the same as that of $3^4.$

Since $3^4$ ends in 1, we can write: . $3^4 \to 1$

Raise both sides to the 25th power: . $\left(3^4\right)^{25} \to 1^{25}$

. . and we have: . $3^{100} \to 1$

• Dec 18th 2008, 06:28 PM
eri
Cut n Paste didn't work, oops
Hi soroban, and thanks. i didn't see that the stuff i cut n paste from the problem didn't work.
So, Most of that was the actual "problem". I wasn't answering my own question, the part after my question was what i was given. sorry for not being more explicit.

I wasn't understanding why they chose 4 as the multiple of 100. because 5 is also a multiple of 100. (3^5)^20 would result in (3^100) as well, or (3)^20...
I see what you did, with the 4, but why is it that 5 couldn't work as well?

thanks so much!!

Quote:

Originally Posted by Soroban
Hello, eri!

You should indicate your exponents more carefully.
Since $3^4$ ends in 1, we can write: . $3^4 \to 1$
Raise both sides to the 25th power: . $\left(3^4\right)^{25} \to 1^{25}$
. . and we have: . $3^{100} \to 1$