perplexus.info :: Just Math : The Triangle's Bottom

The Triangle's Bottom (Posted on 2008-02-05)

Let T be the set of triangular numbers and T* be the set of all products of any two triangular numbers. Show that:

1. Among elements of T, each of the digits 0,1,5 and 6 occur in the units place twice as frequently as each of the digits 3 and 8. (More precisely, if MBk is the set of elements of T that are less than B and end in k, then, e.g., MB1/MB8 approaches 2 as B approaches infinity.)

2. None of the elements of T* end in 2 or 7.

Submitted by FrankM

Rating: 3.0000 (1 votes)

Solution: (Hide)

Problem 1.
The triangular numbers are given by the expression N(N+1)/2 for N= 1, 2, ... It is easy to verify that:
*If N ends in 1,3,6 or 8 then the units digit of the corresponding triangular numbers alternates between 1 and 6
*If N ends in 2 or 7 then the units digit of the corresponding triangular numbers alternate between 3 and 8.
*If N ends in 0,4,5 or 9 then the units digit of the corresponding triangular numbers alternate between 0 and 5.
Hence the frequencies are: 20% each for 0,1,5,6 and 10% each for 3 and 8
Problem 2.
As demonstrated in problem 1, the triangular numbers end in one of the digits 0,1,3,5,6,8. By multiplying these in various combinations, we get numbers ending in 0,1,3,4,5,6,8,9 but it is impossible to get any number ending in 2 or 7.

Comments: ( You must be logged in to post comments.)

Subject	Author	Date
Puzzle Thoughts	K Sengupta	2024-04-09 10:16:09
Can I use bank details to accept money via Cash app number?	Adam Walker	2020-05-04 05:06:06
Is private to make a call on Cash app number?	Adam Walker	2020-05-04 05:04:55
Did it in your sleep..	FrankM	2008-02-06 08:36:11
Solution	Dej Mar	2008-02-06 02:11:50
Solution	Praneeth	2008-02-05 22:53:46
solution	Charlie	2008-02-05 14:48:22

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