I claim that Helen has a list that is 13 more in total than Phil’s.

First of all, we know that Phil almost always has a remainder that is 1 more than Helen.

For example, if we take 7, Helen gets 365-364=1 added to her total while Phil gets 366-364=2 added to his total.

But Phil does have an advantage.
If Helen divides 365 by a number n and receives n-1 as the remainder, we know that Phil would get a nice clean 0 as his remainder.

So the only situations where Phil has an advantage is when:
365 \equiv n-1 \mod n

This simplifies to
366 \equiv 0 \mod n

Thus n must be a factor of 366 or in other words, one of the seven numbers:
1, 2, 3, 6, 61, 122, 183

(Note that we remove 366 simply because it’s bigger than 365)

So time to calculate the difference between Helen and Phil’s lists.
For 365-7 = 358 numbers, Phil gets 1 more than Helen.

But for each of the seven proper divisors n_i of 366, Helen receives n_i-1 more than Phil, who receives 0.

Thus in total, Helen receives 0+1+2+5+121+182=371 more than Phil.

Subtracting 358 from that quantity we get 13, and thus Helen’s list is greater than Phil’s by 13.


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