Find the smallest number that can be subtracted from 148109326 so that it becomes divisible by 8.
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Finding the Smallest Number to Subtract for Divisibility by 8
The question asks for the smallest number that must be subtracted from 148109326 so that the resulting number is divisible by 8. To solve this, we need to understand the divisibility rule for the number 8.
Understanding the Divisibility Rule for 8
A key rule in number theory states that a number is divisible by 8 if and only if the number formed by its last three digits is divisible by 8.
This rule simplifies the problem significantly, as we don't need to consider the entire large number 148109326. We only need to look at its last three digits.
Applying the Divisibility Rule to 148109326
The last three digits of the number 148109326 are 326.
According to the rule, the divisibility of 148109326 by 8 depends entirely on the divisibility of 326 by 8.
To find the smallest number that needs to be subtracted from 148109326 to make it divisible by 8, we need to find the smallest number that, when subtracted from 326, makes the result divisible by 8. This smallest number is the remainder when 326 is divided by 8.
Calculating the Remainder
Let's perform the division of 326 by 8:
We can divide 326 by 8 using standard division methods.
$\frac{326}{8}$
We can think: $8 \times 40 = 320$.
So, $326 = 320 + 6$.
$326 = (8 \times 40) + 6$
In this equation, 40 is the quotient and 6 is the remainder.
This calculation shows that when 326 is divided by 8, the remainder is 6.
Conclusion: Smallest Number to Subtract
The remainder of the division of 326 by 8 is 6. This means that 326 is 6 more than the nearest multiple of 8 that is less than 326 (which is 320).
Therefore, to make 326 divisible by 8, we need to subtract 6 from it ($326 - 6 = 320$, and $320 \div 8 = 40$).
Since the divisibility of 148109326 by 8 depends only on its last three digits (326), subtracting 6 from the entire number 148109326 will result in a number whose last three digits are $326 - 6 = 320$. As 320 is divisible by 8, the resulting large number will also be divisible by 8.
The smallest number that can be subtracted from 148109326 to make it divisible by 8 is the remainder, which is 6.
Let's check the options:
| Option | Value | Check |
|---|---|---|
| 1 | 10 | $326 - 10 = 316$. $316 \div 8 \approx 39.5$. Not divisible by 8. |
| 2 | 8 | $326 - 8 = 318$. $318 \div 8 \approx 39.75$. Not divisible by 8. |
| 3 | 4 | $326 - 4 = 322$. $322 \div 8 \approx 40.25$. Not divisible by 8. |
| 4 | 6 | $326 - 6 = 320$. $320 \div 8 = 40$. Divisible by 8. |
Subtracting 6 results in the number 148109320, which is divisible by 8.
The smallest number to subtract is 6.
Revision Table: Divisibility by 8
| Concept | Explanation |
|---|---|
| Divisibility by 8 Rule | A whole number is divisible by 8 if the number formed by its last three digits is divisible by 8. |
| How to find the smallest number to subtract | Find the remainder when the number (or its last three digits) is divided by 8. The remainder is the smallest number that must be subtracted. |
| Example with 326 | $326 \div 8$ gives a remainder of 6. Subtracting 6 makes it $320$, which is divisible by 8. |
Additional Information: Other Divisibility Rules
Understanding divisibility rules for different numbers can help solve similar problems quickly. Here are a few common ones:
- Divisibility by 2: A number is divisible by 2 if its last digit is even (0, 2, 4, 6, 8).
- Divisibility by 3: A number is divisible by 3 if the sum of its digits is divisible by 3.
- Divisibility by 4: A number is divisible by 4 if the number formed by its last two digits is divisible by 4.
- Divisibility by 5: A number is divisible by 5 if its last digit is 0 or 5.
- Divisibility by 6: A number is divisible by 6 if it is divisible by both 2 and 3.
- Divisibility by 9: A number is divisible by 9 if the sum of its digits is divisible by 9.
- Divisibility by 10: A number is divisible by 10 if its last digit is 0.
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