4 b Is 67 108 divisible by 8? Give a reason for your answer. no 67108 cannot be divisible The number 3812* is divisible by 3. The final digit is missing. What can you say about the missing digit?
4 b Is 67 108 divisible by 8? Give a reason for your answer. no 67108 cannot be divisible The number 3812* is divisible by 3. The final digit is missing. What can you say about the missing digit?
The Correct Answer and Explanation is:
Let’s answer both parts of the question correctly and explain them in detail:
Question 1: Is 67,108 divisible by 8?
Answer:
Yes, 67,108 is divisible by 8.
Reason:
To check if a number is divisible by 8, we look at its last three digits. If the last three digits form a number that is divisible by 8, then the whole number is divisible by 8.
In 67,108, the last three digits are 108.
Now check if 108 ÷ 8 = 13.5 — This is not a whole number, so 108 is not divisible by 8.
Thus, 67,108 is NOT divisible by 8.
Question 2: The number 3812 is divisible by 3. The final digit is missing. What can you say about the missing digit?*
Answer:
To determine the missing digit, we use the rule of divisibility for 3:
A number is divisible by 3 if the sum of its digits is divisible by 3.
We are given 3812*. Let’s add the known digits:
3 + 8 + 1 + 2 = 14
Let the missing digit be x.
So the total sum of the digits becomes: 14 + x
We want 14 + x to be divisible by 3.
Let’s test values of x from 0 to 9:
- 14 + 1 = 15 → Not divisible by 3
- 14 + 2 = 16 → Not divisible
- 14 + 3 = 17 → Not divisible
- 14 + 4 = 18 → Divisible by 3
- 14 + 5 = 19 → Not divisible
- 14 + 6 = 20 → Not divisible
- 14 + 7 = 21 → Divisible by 3
- 14 + 8 = 22 → Not divisible
- 14 + 9 = 23 → Not divisible
- 14 + 0 = 14 → Not divisible
So the values of x that make 3812x divisible by 3 are:
x = 4 or x = 7
✨ Explanation
This problem explores the rules of divisibility, which are quick mental math shortcuts to determine whether a number can be divided evenly by another number without a remainder. These rules are useful in many math problems and exams.
First, for divisibility by 8, the rule is to look only at the last three digits of the number. This is because 1,000 is divisible by 8, so anything before the last three digits won’t affect the divisibility by 8. In the number 67,108, the last three digits are 108. When 108 is divided by 8, it gives 13.5, which is not a whole number. This means 108 is not divisible by 8, and therefore, neither is 67,108. The original statement that “no, 67108 cannot be divisible” is correct, but the explanation should specifically mention checking the last three digits.
Next, for divisibility by 3, we use a different rule: A number is divisible by 3 if the sum of its digits is divisible by 3. In this case, we are given the number 3812*, with the last digit missing. By adding the known digits (3 + 8 + 1 + 2 = 14), we can find which digit should replace the asterisk (*) so that the total sum is divisible by 3. Adding different digits from 0 to 9, we find that adding 4 or 7 gives sums (18 and 21, respectively) that are divisible by 3. So the missing digit must be 4 or 7.
Understanding and applying these simple divisibility rules makes solving such problems much quicker and more efficient.
