Step 1: Understand the Divisibility Rule for 4

A natural number is divisible by $4$ if and only if the number formed by its last two digits (tens and units digits) is divisible by $4$.

$\text{Last two digits} \pmod 4 = 0$

Step 2: Check Option A ($9876754$)

The last two digits are $54$. Dividing $54$ by $4$: $54 \div 4 = 13.5$ (not an integer). Therefore, $9876754$ is not divisible by $4$.

$54 \div 4 \neq \text{integer}$

Step 3: Check Option B ($8978624$)

The last two digits are $24$. Dividing $24$ by $4$: $24 \div 4 = 6$, which is an integer. Thus, $8978624$ is divisible by $4$.

$24 \div 4 = 6$

Step 4: Check Options C and D

For Option C ($7864534$), the last two digits are $34$ ($34 \div 4 = 8.5$). For Option D ($9876342$), the last two digits are $42$ ($42 \div 4 = 10.5$). Neither is divisible by $4$.

$34 \text{ and } 42 \text{ are not multiples of } 4$