To each element of the set S = {1, 2, ..., 1000} a colour is assigned. Suppose that for any two elements a, b of S, if 15 divides (a + b) then they are both assigned the same colour. What is the maximum possible number of distinct colours used?
Answer (2)
The maximum number of distinct colours that can be assigned to the elements of the set S = { 1 , 2 , … , 1000 } under the given condition is 8. This is based on the pairing of elements with complementary remainders when divided by 15. Each of the 8 groups corresponds to a unique colour assignment.
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To determine the maximum possible number of distinct colors that can be used to assign to each element of the set S = { 1 , 2 , … , 1000 } under the condition that for any two elements a , b of S , if 15 divides ( a + b ) , then they must be the same color, we need to analyze the problem using modular arithmetic.
Firstly, let's examine the condition that 15 divides ( a + b ) . For this to happen, ( a + b ) must be a multiple of 15 . This means that a and b have congruent residues modulo 15 , i.e., a ≡ − b ( mod 15 ) , which implies that a and − b are in the same residue class modulo 15 .
In other words, if any two numbers have sums that are multiples of 15 , those numbers must be assigned the same color.
Now, every integer can be expressed in terms of its residue class modulo 15 . The possible residue classes are 0 , 1 , 2 , … , 14 . These 15 residue classes indicate that elements with the same residue modulo 15 must share the same color.
Thus, the number of different residue classes determines the maximum number of distinct colors needed to satisfy the given condition.
Therefore, the maximum number of distinct colors that can be used is 15 , corresponding to the possible distinct residues modulo 15 . This ensures that any pair of numbers adding up to a multiple of 15 (which would share the same color) are properly taken into account by each distinct residue bucket.
