# Number Pattern

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

$5^3=125$

$5^4=625$

$5^5=3125$

$5^6=15625$

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

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

Consider the pattern.

$5^3=125$

$5^4=625$

$5^5=3125$

$5^6=15625$

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

How can I work this out?

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

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

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

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: