In the 100 natural numbers of 1, 2, 3,..., 100, how many different ways are there to take two different numbers so that their sum is a multiple of 7?

In the 100 natural numbers of 1, 2, 3,..., 100, how many different ways are there to take two different numbers so that their sum is a multiple of 7?

Take the sum of two numbers as a multiple of 7. Then consider the remainder of 7
1,2,3,4,5,6,7
8,9,10,11.
.
92 93 94 95 96 97 98
99 100
There are 15 rows in total. There are 7 in the first 14 rows and 2 in the last row
therefore
The remaining 1 = 15
The remaining 2 = 15
The remaining 3 = 14
The remaining 4 = 14
The remaining 5 = 14
The remaining 6 = 14
Integer division = 14
So there are some ways to take any two numbers and a multiple of 7
(1) Take any two numbers in the division: 14 * 13 / 2 = 91
(2) One for each of the remaining 1 and 6 Li: 15 * 14 = 210
(3) Take one from each of the remaining 2 and 5 Li: 15 * 14 = 210
(4) Take one from each of the remaining 3 and 4 Li: 14 * 14 = 196
So there are 91 + 210 + 210 + 196 = 707 methods
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