All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars    
perplexus dot info

Home > Just Math
Oodles of Factors (Posted on 2005-07-08) Difficulty: 2 of 5
A. What is the lowest number that has exactly 10 distinct positive factors?

B. Exactly 1,000 distinct positive factors?

C. Exactly 1,000,000 distinct positive factors?

Example: The distinct factors of 72 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72. Thus 72 has 12 distinct factors.

  Submitted by Leming    
Rating: 3.1250 (8 votes)
Solution: (Hide)
*** Josh and Charlie discovered a better solution ***

A. 48

B. 810,810,000

C. 1.738 x 10^26 (not 2.010 x 10^26)

The quantity of factors for a number can be found by breaking it into its prime factors.

72 = 2 x 2 x 2 x 3 x 3 = 2^3 x 3^2.

2^3 = 8. Now 8 has 4 distinct factors {1, 2, 4, 8} (one more than the exponent)

3^2 = 9 and 9 has 3 distinct factors {1, 3, 9} (one more than the exponent)

By multiplying the two sets of factors together, you get all of the factors for 72. {(1, 2, 4, 8), (3, 6, 12, 24), (9, 18, 36, 72)}

By multiplying the quantity of distinct factors, you get 3 x 4 = 12 distinct factors.

A. Reversing this process, for a number with 10 distinct factors:

10 = 5 x 2

Using the lowest 2 primes {2, 3} and the factors of 10, the following equation results:

2^(5-1) x 3^(2-1)

= 2^4 x 3 = 48

So the answer to Part A is 48.

B. 1000 distinct factors:

1000 = 5 x 5 x 5 x 2 x 2 x 2

First 6 primes {2, 3, 5, 7, 11, 13)

2^(5 – 1) x 3^(5 – 1) x 5^(5 – 1) x 7^(2 – 1) x 11^(2 – 1) x 13^(2 – 1)

= 2^4 x 3^4 x 5^4 x 7 x 11 x 13

= 810,810,000

C. 1,000,000 distinct factors

1,000,000 = 5 x 5 x 5 x 5 x 5 x 5 x 2 x 2 x 2 x 2 x 2 x 2

First 12 primes {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37}

2^4 x 3^4 x 5^4 x 7^4 x 11^4 x 13^4 x 17 x 19 x 23 x 29 x 31 x 37

= 2320^4 x 247,110,827

= 200,961,610,708,938,459,249,870,000

Correct answer from Josh and Charlie redistributed the factoring (2^9 vice 2^4 x 37) to get:

= 173,804,636,288,811,640,432,320,000

Comments: ( You must be logged in to post comments.)
  Subject Author Date
Puzzle Thoughts K Sengupta2023-05-21 02:23:57
The answer to A is...Ed2005-07-15 21:40:01
computer exhaustive searchCharlie2005-07-08 20:34:02
re: general(ish) sol'n (w/o computer) -correctionJosh706792005-07-08 20:21:29
Solutiongeneral(ish) sol'n (w/o computer)Josh706792005-07-08 20:06:17
Solutionre: No SubjectCharlie2005-07-08 19:26:44
Hints/TipsNo SubjectCharlie2005-07-08 16:20:59
Some Thoughtspart A solution?john2005-07-08 15:36:33
Please log in:
Login:
Password:
Remember me:
Sign up! | Forgot password


Search:
Search body:
Forums (0)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (14)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Chatterbox:
Copyright © 2002 - 2024 by Animus Pactum Consulting. All rights reserved. Privacy Information