Question

# A number n! is written in base 6 and base 8 notation. Its base 6 representation ends with 10 zeroes. Its base 8 representation ends with 7 zeroes. Find the smallest n that satisfies these conditions. Also find the number of values of n that will satisfy these conditions

CAT 2021

A
B
C
D
##### 25 and 4
Solution

Correct option is

(C)

According to ques , the number is a multiple of $6^{10}$

if n! is a multiple of $6^{10}$ , it has to be a multiple of $3^{10}$

the smallest factorial should be 24! , so , when n= 24, 25 or 26 , n! will be multiple of $6^{10}$ (not $6^{11}$)

similarly ,we have to find n! such that it is a multiple of $2^{21}$

but not $2^{24}$ . when n = 24 , 25 , 26 , 27 , n! will be multiple of $2^{21}$ but not $2^{24}$

hence the smallest n from 24 , 25 , 0r 26 , will be 24 .

hence C is the correct option , "24 , 3"
{"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": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 9, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8877742886543274, "perplexity": 409.8837150787312}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "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": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030335514.65/warc/CC-MAIN-20221001003954-20221001033954-00352.warc.gz"}