Question

To each element of set S=(1,2,3..........1000)a color is assigned .suppose that for any two elements a,b of S ,if 15 divides (a+b)then they are both same colour .what is the maximun posible numers of distinct colour used?

Answer 1

Answer:

8

Step-by-step explanation:

To each element of set S=(1,2,3..........1000)a color is assigned .suppose that for any two elements a,b of S ,if 15 divides (a+b)then they are both same colour .what is the maximun posible numers of distinct colour used?

Unique combinations :

1 + 14    (1 + 15n  & 14 + 15n) = 15 + 15k

2 + 13   (2+ 15n  & 13 + 15n) = 15 + 15k

3 + 12   (3 + 15n  & 12 + 15n)  = 15 + 15k

4 + 11    (4 + 15n  & 11 + 15n)  = 15 + 15k

5 + 10   (5 + 15n  & 10 + 15n) = 15 + 15k

6 + 9     (6 + 15n  & 9 + 15n) = 15 + 15k

7  + 8    (7 + 15n  & 8 + 15n) = 15 + 15k

15 + 30   (15n + 15n) =  15k

the maximun possible numbers of distinct colour used = 8

Answer 2

Answer:

Let's try .

For numbers from 1 to 15:

1+14=15

2+13=15

3+12=15

4+11=15

5+10=15

6+9= 15

7+8=15

IN ALL THESE SEVEN CASES a+b are exactly divisible by 15 . So these first 14 numbers would contribute to 7 distinct colours.

Let us now assign 15 a different colour(it is not making any a+b/15=Z combination with the first 14 numbers, so it's colour can't be the same as theirs )

Now we have a total of 8 different colours.

Now, all the numbers from 16 to 100 would require 15 or less than it to reach a multiple of 15 , that is , numbers from 1 to 15 . This way, they can be proved equal to the previous numbers and the total number of colours used becomes equal to 8.

Therefore answer = 8