$1, 2, 3, 4 ,\ldots, 300$, from this series, you need to erase some numbers and create a new series so that any two numbers’ sum is not divisible by $7$. What is the number of maximum terms that could be in the new series?
I guess some are confused regarding the wording of this question. It was originally written in another language and I've simply translated it. The questions asks to make a new sequence from the above sequence which is $1,2,3,4, \ldots,300$. You will remove numbers from this sequence such that if you choose any two numbers from your new sequence, and them add them up, this summation should NOT be divisible by $7$. You'll have to find the maximum numbers of terms of your new sequence.
How do I approach such a problem?
I've come to a part in this problem where I've found out that numbers that add up to $7$'s multiples should not be included in the sequence. For example, in the case of $13$ and $1$, they add up to $14$, which is divisible by $7$. So both these numbers can't be included in this new sequence.
If I consider the number $1$, I can add it to the following sequence: $6, 13, 20, 27,\ldots$ and I would get multiples of $7$. So this sequence can't be in my new sequence. This also goes for $5, 12, 19, 26,\ldots$ when added with $2$. Similarly, $4, 11, 18, 25, \ldots$ with $3$. In all the above sequences, here are $42$ terms, so I thought of subtracting $3$ times of $42$ from the initial $300$ terms to get the maximum number of terms. However, that method actually turned out to be wrong.