# Let $n$ be the positive integer such that $5^n$ and $2^n$ begin with same digit. Which digit is that?

This is my first time posting here so sorry if I done something wrong, and also my first time encountering a problem like this.

Besides trivial $0$, the only solution I found by simply writing down the powers of $2$ and $5$ parallely, is $5$ ($5^5=3125$, $2^5=32$). I couldn't find any kind of period. I've done problems which are about last digit, but not about the first one.

Hopefully I could get a hint. Thanks.

-

## 1 Answer

Hint: $5^n\times2^n=10^n$. ${}$

-

2
+1 This is just so... I'm gonna use this as an exercise given half a chance. – Jyrki Lahtonen Feb 6 '12 at 12:04
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 11, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9400873780250549, "perplexity_flag": "head"}