Posted by **Krish** on Friday, December 27, 2013 at 4:10pm.

How many ordered pairs of natural numbers (n, m) are there that solve the equation

2n + 3m = 100

- Math -
**Damon**, Friday, December 27, 2013 at 5:02pm
Well we could argue about if zero counts as a natural number. If it does then

2n will always be zero or even

2 n = 0 2 4 6 8 ...... 50 ....100

3 m is zero or a multiple of 3

3 m = 0 3 6 9 12 15 ....33 .... 99

now the sum of those must be 100, which is even so delete all the odd 3ms

3 m = 0 6 12 18 24 .... 90 96

now which could add to 100

4 + 96

10 + 90

16 + 84

22 + 78

28 + 72 add six, subtract 6, see pattern

34 + 66

40 + 60

46 + 54

52 + 48

58 + 42

64 + 36

70 + 30

76 + 24

82 + 18

88 + 12

94 + 6

100 + 0 count them :)

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