To solve for $d$ such that $4 \mid 84546d$, we need to apply the divisibility rule for 4. A number is divisible by 4 if the number formed by its last two digits is divisible by 4.

The number is $84546d$, where $d$ is the unknown digit.

Step 1: Focus on the last two digits

The last two digits of the number $84546d$ are $6d$ (since $d$ is in the ones place).

For divisibility by 4, we need $6d$ (the number formed by 6 and $d$) to be divisible by 4.

Step 2: Check the values of $d$

Now we check the possible values for $d$ (ranging from 0 to 9) to see which make $6d$ divisible by 4.

• For $d = 0$, $6d = 60$ → $60 \div 4 = 15$ (divisible by 4)
• For $d = 1$, $6d = 61$ → $61 \div 4 = 15.25$ (not divisible by 4)
• For $d = 2$, $6d = 62$ → $62 \div 4 = 15.5$ (not divisible by 4)
• For $d = 3$, $6d = 63$ → $63 \div 4 = 15.75$ (not divisible by 4)
• For $d = 4$, $6d = 64$ → $64 \div 4 = 16$ (divisible by 4)
• For $d = 5$, $6d = 65$ → $65 \div 4 = 16.25$ (not divisible by 4)
• For $d = 6$, $6d = 66$ → $66 \div 4 = 16.5$ (not divisible by 4)
• For $d = 7$, $6d = 67$ → $67 \div 4 = 16.75$ (not divisible by 4)
• For $d = 8$, $6d = 68$ → $68 \div 4 = 17$ (divisible by 4)
• For $d = 9$, $6d = 69$ → $69 \div 4 = 17.25$ (not divisible by 4)

Step 3: Conclusion

The values of $d$ that make $84546d$ divisible by 4 are $d = 0, 4, 8$.

Would you like further explanation or details on this solution?

Here are some follow-up questions:

1. Can you explain the divisibility rule for 4 more thoroughly?
2. How would this work with divisibility rules for other numbers, like 3 or 5?
3. Can you solve the same type of problem for divisibility by 6?
4. How would the answer change if we used a different number with more digits?
5. What is the importance of divisibility rules in number theory?

Tip: Divisibility rules can help solve many number theory problems quickly without needing to perform full division.