MAH-CET 2026 — Mathematics PYQ

To find the sum of integers divisible by $2$ or $5$, we use the Principle of Inclusion-Exclusion from Set Theory.

The formula for the total sum is:

$$\text{Total Sum}=(\text{Sum of numbers divisible by }2)+(\text{Sum of numbers divisible by }5)-(\text{Sum of numbers divisible by both }2\text{ and }5)$$

Numbers divisible by both $2$ and $5$ are the numbers divisible by their Least Common Multiple (LCM), which is $10$.

Let us calculate each sum individually using the Arithmetic Progression (AP) sum formula:

$$S_n=\frac{n}{2}[a+l]$$

Where:

• $n$ = number of terms

• $a$ = first term

• $l$ = last term

Step 1: Sum of numbers divisible by $2$ ($S_2$)

The sequence of numbers from $1$ to $100$ divisible by $2$ is:

$$2,4,6,8,\dots ,100$$

Here:

• First term ($a$) = $2$

• Last term ($l$) = $100$

• Number of terms ($n$) = $\frac{100}{2}=50$

Calculating the sum ($S_2$):

$$S_2=\frac{50}{2}[2+100]$$

$$S_2=25\times 102=2550$$

Step 2: Sum of numbers divisible by $5$ ($S_5$)

The sequence of numbers from $1$ to $100$ divisible by $5$ is:

$$5,10,15,20,\dots ,100$$

Here:

• First term ($a$) = $5$

• Last term ($l$) = $100$

• Number of terms ($n$) = $\frac{100}{5}=20$

Calculating the sum ($S_5$):

$$S_5=\frac{20}{2}[5+100]$$

$$S_5=10\times 105=1050$$

Step 3: Sum of numbers divisible by $10$ ($S_{10}$)

The sequence of numbers from $1$ to $100$ divisible by both $2$ and $5$ (i.e., divisible by $10$) is:

$$10,20,30,40,\dots ,100$$

Here:

• First term ($a$) = $10$

• Last term ($l$) = $100$

• Number of terms ($n$) = $\frac{100}{10}=10$

Calculating the sum ($S_{10}$):

$$S_{10}=\frac{10}{2}[10+100]$$

$$S_{10}=5\times 110=550$$

Step 4: Combine the sums using the Principle of Inclusion-Exclusion

Now, substitute the values back into our main formula:

$$\text{Required Sum}=S_2+S_5-S_{10}$$

$$\text{Required Sum}=2550+1050-550$$

$$\text{Required Sum}=3600-550$$

$$\text{Required Sum}=3050$$

Conclusion

The sum of integers from $1$ to $100$ which are divisible by $2$ or $5$ is $3050$.

Hence, the correct option is (b) 3050.