Pairs with Sum divisible by 5
Python 2, 32 bytes
lambda m,n:(m*n+abs(m%5-~n%5))/5
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Adapts an idea from Arnauld. We estimate that \$1/5\$ of the \$mn\$ pairs will satisfy the mod-5 condition. Th actual value may be one higher depending on the mod-5 values of \$m\$ and \$n\$, and we bump up \$mn\$ a bit so that these make it pass the next multiple of 5.
34 bytes
lambda m,n:m*n/5+(0<-m%5*(-n%5)<5)
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34 bytes
lambda m,n:m*n/5+(m*n%5+m%5+n%5)/9
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36 bytes
lambda m,n:m*n/5+((5-m%5)*(5-n%5)<5)
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36 bytes
lambda m,n:m*n/5+(abs(m%5+n%5*1j)>4)
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36 bytes
lambda m,n:m*n/5+(m%5*5/4+n%5*5/4>5)
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36 bytes
lambda m,n:m*n/5+(m%5+n%5+m*n%5/4>5)
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JavaScript (ES6), 29 bytes
Takes input as (m)(n).
m=>g=n=>n&&(m+n%5)/5+g(n-1)|0
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How?
Let's consider a grid of width \$m\$ with 1-indexed coordinates and an X mark on each cell for which:
$$(x+y)\equiv 0\pmod 5$$
Here are the first few rows for \$m=9\$:
| 1 2 3 4 5 6 7 8 9
---+-------------------
1 | - - - X - - - - X
2 | - - X - - - - X -
3 | - X - - - - X - -
4 | X - - - - X - - -
5 | - - - - X - - - -
6 | - - - X - - - - X
7 | - - X - - - - X -
Computing \$m+(y\bmod5)\$ is equivalent to left-padding each row in such a way that all X marks are vertically aligned and appear on columns whose index is a multiple of \$5\$.
With such a configuration, the number of marks is directly given by \$\lfloor{L_y/5}\rfloor\$, where \$L_y\$ is the updated length of the \$y\$-th row.
y | y%5 | padded row | length | // 5
---+-----+----------------------------+--------+------
1 | 1 | + - - - X - - - - X | 10 | 2
2 | 2 | + + - - X - - - - X - | 11 | 2
3 | 3 | + + + - X - - - - X - - | 12 | 2
4 | 4 | + + + + X - - - - X - - - | 13 | 2
5 | 0 | - - - - X - - - - | 9 | 1
6 | 1 | + - - - X - - - - X | 10 | 2
7 | 2 | + + - - X - - - - X - | 11 | 2
---+-----+----------^---------^-------+--------+------
| 0 0 0 0 0 0 0 0 0 1 1 1 1 |
| 1 2 3 4 5 6 7 8 9 0 1 2 3 |We use this method to recursively compute the number of marks on each row.
Jelly, 5 bytes
p§5ḍS
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How?
p§5ḍS - Link: positive integer, M; positive integer N
p - (implicit [1..M]) Cartesian product (implicit [1..N])
§ - sums
5ḍ - five divides? (vectorises)
S - sum