# Number Pattern

• Sep 19th 2009, 11:02 PM
Number Pattern
Consider the pattern.

$\displaystyle 5^3=125$

$\displaystyle 5^4=625$

$\displaystyle 5^5=3125$

$\displaystyle 5^6=15625$

Use this pattern to write out the last 3 digits of $\displaystyle 5^{15}$.

How can I work this out?
• Sep 19th 2009, 11:13 PM
CaptainBlack
Quote:

Consider the pattern.

$\displaystyle 5^3=125$

$\displaystyle 5^4=625$

$\displaystyle 5^5=3125$

$\displaystyle 5^6=15625$

Use this pattern to write out the last digits of $\displaystyle 5^{15}$.

How can I work this out?

1. You do not specify how many digits constitute "the last" digits of $\displaystyle 5^{15}$.

2. A suitable hypothesis might be that the last two digits of $\displaystyle 5^n$ for $\displaystyle n\ge 2$ are $\displaystyle 25$.

3. Suppose this true for $\displaystyle k$ that is $\displaystyle 5^k=100 \times M+25$ for some integer $\displaystyle M$, now $\displaystyle 5^{k+1}= ...$

NOw can you take it from there?

CB
• Sep 19th 2009, 11:51 PM
Sorry, it was the last three digits. This was easier than I thought! How silly of me!!

Seeing as odd indices always result in ending digits of 125, and even indices result in 625, the answer to this problem would be 125. Am I right?
• Sep 20th 2009, 01:48 AM
CaptainBlack
Quote: