find the smallest natural number which when added to the difference of square of 17 and 13 gives a perfect square.
the difference between 17^2 and 13^2 is 120.
the next perfect square is 121
So we have to add 1
To find the smallest natural number that satisfies the given condition, let's break down the problem step-by-step.
1. First, find the difference of the square of 17 and 13:
Difference = 17^2 - 13^2 = 289 - 169 = 120
2. Now, we need to find the smallest natural number that, when added to 120, gives a perfect square.
To solve this, let's find the factors of 120 and see if any of them, when added to 120, gives a perfect square:
Factor pairs of 120:
1, 120
2, 60
3, 40
4, 30
5, 24
6, 20
8, 15
10, 12
Testing each factor pair:
1 + 120 = 121 (a perfect square)
2 + 60 = 62 (not a perfect square)
3 + 40 = 43 (not a perfect square)
4 + 30 = 34 (not a perfect square)
5 + 24 = 29 (not a perfect square)
6 + 20 = 26 (not a perfect square)
8 + 15 = 23 (not a perfect square)
10 + 12 = 22 (not a perfect square)
Therefore, the smallest natural number that, when added to the difference of the square of 17 and 13, gives a perfect square is 1.