Increase in number of factors. (Posted on 2010-10-05)

	Submitted by Vishal Gupta
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General formula for number of factors of (a^x X b^y X c^z) is (x+1)(y+1)(z+1). If n has 15 factors, it can either be a^14 or a^4 X b^2. If a^14 then 2n will be either 2^15 (16 factors) or 2^1 X a^14 (30 factors). If a^4 X b^2 then 2n can be (i) 2^1 X a^4 X b^2 (30 factors) (ii) 2^5 X b^2 (18 factors) (iii) a^4 X 2^3 (20 factors) So, this has to be the case. This means original number was a^4 X 2^2. Then 4n will be a^4 X 2^4 (25 factors). So 4n will definitely have 25 factors. 5n can be either (i) a^4 X 2^2 X 5^1 (30 factors) (ii) 5^5 X 2^2 (18 factors) So 5n can have either 18 or 30 factors depending on whether the original number had a power of 5 or not. Other solutions can be found in the comments.

	Subject	Author	Date
	Puzzle Answer	K Sengupta	2022-07-10 21:26:19
	4n will have precisely 25 factors.	K Sengupta	2022-07-10 21:24:16
	solution	Charlie	2010-10-05 17:22:19
	Solution	Jer	2010-10-05 14:55:03
	Possible solution	broll	2010-10-05 14:27:58