(b) The sum of integers from $1$ to $100$ that are divisible by $2$ or $5 =$ sum of series divisible by $2 +$ sum of series divisible by $5 -$ sum of series divisible by $2$ and $5$.

$ = (2 + 4 + 6 + …… + 100) + (5 + 10 + 15……. + 100)$

$ – (10 + 20 + 30 + …….. + 100)$

$ = \frac{{50}}{2}\left\{ {2 \times 2 + (50 – 1)2} \right\} + \frac{{20}}{2}\left\{ {2 \times 5 + (20 – 1)5} \right\}$

$ – \frac{{10}}{2}\left\{ {10 \times 2 + (10 – 1)10} \right\}$

$ = 2550 + 1050 – 550 = 3050$.