Tag Archives: natural

12. Trigangle Numbers

The sequence of triangle numbers is generated by adding the natural numbers. So the 7th 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: 1
2 => 3: 1,3
3 => 6: 1,2,3,6
4 => 10: 1,2,5,10
5 => 15: 1,3,5,15
6 => 21: 1,3,7,21
7 => 28: 1,2,4,7,14,28 = 6
8 => 36= 1,2,3,4,6,9,12,13,16,18 = 10
9 => 45 = 1,3,5,9,15,45 = 6
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? Read more of this post

Advertisement

10. Sum of Prime

The sum of the primes below 10 is 2 + 3 + 5 + 7 = 17.  Find the sum of all the primes below two million.

A prime number (or a prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself.   A natural number greater than 1 that is not a prime number is called a composite number.   For example, 5 is prime, as only 1 and 5 divide it, whereas 6 is composite,   since it has the divisors 2 and 3 in addition to 1 and 6. Read more of this post

Pythagorean triplet

A Pythagorean triplet is a set of three natural numbers, a < b < c, for which,  a2 + b2 = cFor example, 32 + 42 = 9 + 16 = 25 = 52. There exists exactly one Pythagorean triplet for which a + b + c = 1000.  Find the product abc.

Simple approach a2 + b2 = (N-a-b)2   => a+b+c =N   i.e., a<b<c, From the condition a < b < c, we conclude that
a <= (s − 3) /3 and   b < (s − a)/2.

Using a parametrization of Pythagorean triplets

A Pythagorean triplet (a, b, c) is by definition primitive if gcd(a, b, c) = 1. Since for Pythagorean triplets one has gcd(a, b) = gcd(b, c) = gcd(c, a), such a triplet is primitive if and only if gcd(a, b) = 1. Read more of this post

Sum of the squares natural numbers

The sum of the squares of the first ten natural numbers is,  12 + 22 + … + 102 = 385. The square of the sum of the first ten natural numbers is,  (1 + 2 + … + 10)2 = 552 = 3025.  Hence the difference between the sum of the squares of the first ten natural numbers and the square of the sum is 3025 − 385 = 2640.

Find the difference between the sum of the squares of the first one hundred natural numbers and the square of the sum.

sum(n) = n(n + 1)/2
sum(n^2) = (2n+1) (n+1) n/6
Read more of this post

1.Natural Numbers

List all the natural numbers below 10 that are multiples of 3 or 5, we get 3, 5, 6 and 9.  The sum of these multiples is 23. Find the sum of all the multiples of 3 or 5 below 1000.

The term “natural number” refers either to a member of the set of positive integers 1, 2, 3,) or   to the set of non-negative integers 0, 1, 2, 3, ….

set name symbol
…, -2, -1, 0, 1, 2, … integers Z
1, 2, 3, 4, .. positive integers Z-+
0, 1, 2, 3, 4, … nonnegative integers Z-*
0, -1, -2, -3, -4, … nonpositive integers
-1, -2, -3, -4, … negative integers Z–

Read more of this post