 Try this beautiful problem from the Pre-RMO, 2019 based on Natural Numbers.

## Natural numbers Problem – PRMO 2019

Let E denote the set of all natural number n such that $3< n<100$ and the set {1,2,3,…,n} can be partitioned in to 3 subsets with equal sums. Find the number of elements of E.

• is 107
• is 64
• is 840
• cannot be determined from the given information

### Key Concepts

Divisibility

Equations

Integer

But try the problem first…

Source

PRMO, 2019, Question 30

Elementary Number Theory by David Burton

## Try with Hints

First hint

{1,2,…,n}

This set can be partitioned into 3 subsets with equal sums so total sum is divisible by 3

$\frac{n(n+1)}{2}$ is divisible by 3.

or, n of form 3k, 3k+2

or, n=6k,6k+2,6k+3, 6k+5

Second Hint

case I n=6k, we group numbers in bundles of 6 for each bundle 1,2,3,4,5,6(16,25,34)

case II n=6k+2 then we club last bundle of 8 numbers rest can be partitioned and those eight numbers can be done 1,2,3,4,5,6,7,8 (1236,48)

case III n=6k+3 we club last nine number and rest can be partitioned 1,2,3,4,5,6,7,8,9 (12345,69,78)

Final Step

case IV 6k+5 we take last five numbers, rest can be aprtitioned 1,2,3,4,,5(14,25,5)

Hence we select any number of form 6k(16), 6k+2(16), 6K+3(16), 6K+5(16)

or, total=64 numbers.