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When you divide an integer by 6, the remainder could not be which of the following? Select all that apply
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When you divide an integer by 6, the remainder could be 0, 1, 2, 3, 4, or 5.
We already know that the remainder cannot be negative.
Moreover, you cannot have a remainder larger than or equal to 6 (d).
Since you admire analogies, let’s use one to clear this concept.
Let’s revisit a slightly altered version of the oranges problem you so liked in the previous micro-lesson.
Consider the problem “You have 39 oranges, there are 5 people, and each person should get the same number of oranges. What is the minimum number of remaining oranges?”. In mathematical terms, n is 39, and d is 5.
The answer is obviously 4, the remainder of 39 divided by 5. The quotient is 7, meaning each person got 7 oranges and there were 5 people, so a total of 35 oranges got divided amongst these 5 people.
But why is the remainder less than 5? Why can’t it be more?
Well, imagine if 5 or more oranges remain, you can still go ahead and distribute one more orange to each person. So it means 5 or any larger number cannot be the answer. Therefore, the remainder should be less than 5.
Important trick
In remainder problems, the question might trick you with a negative value of d. For example, what is the remainder when you divide an integer by -6?
The answer is the same as that for the positive value of d.
When you divide an integer by -6, the remainder could be 0, 1, 2, 3, 4, or 5.
In other words, the 0 ? Remainder < |-6| ? 0 ? Remainder < 6
Remember |-6| gives us the absolute value or the magnitude of -6, which is 6.
An alternative approach
Let’s say you want to find the remainder of 17 divided by ?6.
To find the remainder, we can also use the 3-step process.
- Step 1: List all the multiples of d (?6) that are less than the n (17). Those are ?6, 0, 6, and 12.
- Step 2: Select the multiple that is nearest to n. In this case that is 12.
- Step 3: Find the distance from the n (17) to the multiple you selected in step 2 (12).
17 – 12 = 5. This is your remainder.
We can see that the remainder (5) is less than 6 (the absolute value of ?6, the divisor).
So, if you have a negative value of d, you can generalize this as:
0 ? Remainder < |d|
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What is the minimum possible remainder when an integer is divided by 11? Since, 0 ? Remainder < d, the minimum possible value of the remainder is 0. In other words, n is completely divided by d. For example, 11 ÷ 11, 22 ÷ 11 etc… |
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What is the minimum possible remainder when an integer is divided by ?11? Recall that if you have a negative value of d, 0 ? Remainder < |d|, Again, the minimum possible value of the remainder is 0. In other words, n is completely divided by d. For example, 11 ÷ ?11, 22 ÷ ?11 etc… |
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And what is the maximum possible remainder when an integer is divided by 11? Since, 0 ? Remainder < d, the value of the remainder should lie somewhere within that range. In this question, the d is 11, so the range of remainder values is 0 ? Remainder < 11 This means that the remainder can be any value starting from 0 and going up to less than 11. 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 As the question is asking about the maximum possible value, the answer would be 10. |
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And what is the maximum possible remainder when an integer is divided by ?11? For a negative d, 0 ? Remainder < |d|, the value of the remainder should lie somewhere within that range. In this question, d is ?11, so the range of remainder values is 0 ? Remainder < |?11| ? 0 ? Remainder < 11 This means that the remainder can be any value starting from 0 and going up to less than 11. 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 As the question is asking about the maximum possible value, the answer would be 10. |
You can also generalize this as:
If d is positive, the maximum possible value of the remainder is 1 less than d.
If d is negative, the maximum possible value of the remainder is 1 less than |d|.
To summarise:
- If d is +ve
0 ? Remainder < d
The maximum possible value of the remainder = d – 1
- If d is -ve
0 ? Remainder < |d|
The maximum possible value of the remainder = |d| – 1