1. Bring the numbers down so that they all become a multiple of 3 and count how many times they have to come down
Ex)
4.5 7 8 9
1 2 1 2. 0
2. We only have to add up these small numbers(1-2-0) because if they add up and make a number which is a multiple of 3, the left over 3s in the number will make a multiple of 3 anyways.
Different situations
1,1,1,1,1=1+1+1=3
2,2,2,2,2=2+2+2=6
0,0,0,0,0=0+0+0=0
1,1,2,2,0=2+1+0=3
2,2,2,2,0=2+2+2=6
2,2,0,0,0=0+0+0=0
1,1,1,1,2=1+1+1=3
These combinations will always work
Possible combinations of 3 numbers that make a multiple of 3 are:
1+1+1
0+0+0
2+2+2
1+2+0
One of these set of numbers will always be in any 5 numbers
If numbers are positive
Ex)
4 7 10 24 59
1 1 1 0 2
4 is 1 apart from 3
7 is 1 apart from 6
10 is 1 apart from 1
24 is 0 apart from 24
59 is 2 apart from 57
3,6,1,24,57 these 5 numbers are multiple of 3 anyways, so we don’t have to count them
1,1,1,0,2
We can add either
1+1+1=3 or 1+0+2=3
If we use the original numbers:
4+7+10=21(1+1+1)
Or
4+24+59=87(1+0+2)
If numbers are negative or negative&positive
Ex)
-5 -6 -19 -12 5
1 0 2 0 2
-5 is 1 away from -6
-6 is 0 away from -6
-19 is 2 away from -21
-12 is 0 away from -12
5 is 2 away from 3
1,0,2,0,2
We could add
1+0+2=3
-5+(-6)+(-19)
-11-19=-30
-30 is a multiple of 3