Question

The sequence of triangle numbers is generated by adding the natural numbers. So the 7^th triangle number would be 1 + 2 + 3 + 4 + 5 + 6 + 7 = 28. The first ten terms would be:

1, 3, 6, 10, 15, 21, 28, 36, 45, 55, ...

Let us list the factors of the first seven triangle numbers:

1: 1
3: 1,3
6: 1,2,3,6
10: 1,2,5,10
15: 1,3,5,15
21: 1,3,7,21
28: 1,2,4,7,14,28

We can see that 28 is the first triangle number to have over five divisors.What is the value of the first triangle number to have over five hundred divisors? Solve using only math.

Answer 1

solution : - we know that triangular nummer are giun by No of divisar E1=1 = 20.30.5 [2 = 3 = 2.31.50 - - - 1,3 E3 = 6 = 21.395 . 112,3, 6 1,2,4,7, 14,28 E7 = 28 = 2239.6°7'. Now, wham to find a Numhur (triangular) which has 500+ diuisars. w Kniw that, By Prime Decomposition thewom. Elwry nin can be writtanas, n = p? q B zu ... where ,piwr are realiny brine and, 2, B, 4 --- are paritum integer Alsoo 500 = 2x2x5x5x5. Now. Consider the numher 576 = 3x3 x 4x2x2x2x2 wolle n = 2² 3² 5 ² 71.11.131. 171 = In = 765 76500 which is the peauired number having our Soo diuisar. 10 576.