The Apple Problem

Number Theory Level 4

There is a box full of apples, it has more than 80 but less than 85 apples in the box, some of the apples are red and the others are green, it is known that the number of red apples can be expressed as \(4k-3\) where \(k\) is an integer.

After a while, a person took away more than 15 green apples and ran away.

Then, another person came and checked the box, he wants to match 1 red apple and 1 green apple as a pair, after he matched all the possible pairs, he finds out there are less than 13 red apples that can't be matched as a pair.

If he switches the red apples the same number as \(\frac{1}{4}\) of stolen green apples to green, then only 2 red apples can't be matched as a pair.

If there are originally at least 37 red apples in the box, what is the original number of apples in the box?


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