If the integers m and n are chosen at random between 1 and 100, then the probability that 7m +7n is divisible by 5 is

# If the integers m and n are chosen at random between 1 and 100, then the probability that 7m +7n is divisible by 5 is

1. A

$\frac{1}{4}$

2. B

$\frac{1}{7}$

3. C

$\frac{1}{8}$

4. D

$\frac{1}{49}$

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### Solution:

${7}^{1}=7,{7}^{2}=49,{7}^{3}=343,{7}^{4}=2401$

$\therefore$ For each value of m in $\left\{1,2,3,\dots ,100\right\}$ there are
25 values of n as follows

$\begin{array}{l}\left(1,3\right),\left(1,7\right),\left(1,11\right),\dots \dots \dots \dots \dots ,\left(1,99\right)\\ \left(2,4\right),\left(2,8\right),\left(2,12\right),\dots \dots \dots \dots \dots ,\left(2,100\right)\\ \left(3,1\right),\left(3,5\right),\left(3,9\right),\dots \dots \dots \dots \dots ,\left(3,97\right)\end{array}$

$\left(4,2\right),\left(4,6\right),\left(4,10\right),\dots \dots \dots \dots \dots ,\left(4,98\right)$ etc.

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