1. We need the sum of all integers from $1$ to $100$ that are divisible by $2$ or $5$.

2. Use the inclusion-exclusion principle:

   • Sum of multiples of $2$
   • Sum of multiples of $5$
   • Subtract sum of multiples of both $2$ and $5$, i.e. multiples of $10$

Step 1: Sum of multiples of $2$ from $1$ to $100$

The multiples are: $2,4,6,\dots,100$ This is an AP with:

• first term $a=2$
• last term $l=100$
• number of terms $n=\frac{100}{2}=50$

So, $S_2=\frac{50}{2}(2+100)=25\cdot 102=2550$

Step 2: Sum of multiples of $5$ from $1$ to $100$

The multiples are: $5,10,15,\dots,100$ Here,

• first term $a=5$
• last term $l=100$
• number of terms $n=\frac{100}{5}=20$

So, $S_5=\frac{20}{2}(5+100)=10\cdot 105=1050$

Step 3: Sum of multiples of both $2$ and $5$

Numbers divisible by both $2$ and $5$ are divisible by $\operatorname{lcm}(2,5)=10$.

Multiples of $10$ from $1$ to $100$ are: $10,20,30,\dots,100$ Here,

• first term $a=10$
• last term $l=100$
• number of terms $n=\frac{100}{10}=10$

So, $S_{10}=\frac{10}{2}(10+100)=5\cdot 110=550$

Step 4: Apply inclusion-exclusion

Required sum: $S=S_2+S_5-S_{10}$ $S=2550+1050-550$ $S=3050$

Step 5: Check options

• A: $3000$
• B: $3050$ ✅
• C: $3600$
• D: $3250$

So the correct option is: $\boxed{\text{B: }3050}$