Grades 5-6 Video Solutions 2015-Level 5&6 Video Solutions 2015 problem15

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Question number 15. A student wrote down a natural number. When she divided that number by 9, the remainder was 7. What is the remainder when twice that number is divided by 9?  We have a division problem here with a number, an unknown number, being divided. We know that the divisor was 9, we don't know the quotient, and we do know that the remainder  was 7. We can relate all of this information to each other using the division algorithm. I'll say that the natural number here that we're talking about is M. We'll use Q for  the quotient, and the other information is no. So M is going to be the natural number, and I'll use Q for the quotient. And so what we have is M is equal to the divisor times  the quotient, so that's 9 times Q, plus the remainder, which is 7. And then with this equation in mind, we're looking for twice the original number, which would be 2 times  M. In my equation, I have to multiply both sides by 2. 2 times M is going to be the product of 2 and 9 and Q, plus now I have twice the remainder here. And so that gives me  18 times Q plus 14. Since the divisor is still 9, we have to make sure that the remainder is  not too large, and here the remainder  is in fact greater than the divisor.  What that means for us performing the division algorithm is we have to keep going, so we keep dividing. And out of this 14, I can pull out a 9, an extra copy of the divisor. So  the next line should look like this, 18 times Q plus, write 14 as 9 plus 5, and then try to  rewrite everything in the original form. So if I factor out that 9, I'll have a 2 times Q plus 1,  and then a plus 5. Now 5 here is less than the divisor of 9. We have a new quotient here, which I'll underline in orange, but we don't really care about that number. The number we  do care about is the remainder, which cannot be reduced anymore, since it is smaller than the divisor. So we are done with the division algorithm. We have divided completely,  and we see that the remainder out of this new division of twice the original number is 5.  That is answer C.

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