**Week 3: Remainders – Day 5**

1. If you divide 87 by 23, what is the quotient and what is the remainder?

The multiples of 23 are 23, 46, 69, 92, and so on. The multiple nearest to and below 87 is 69. The quotient is therefore 3, and the remainder = 87 – 69 = 18

2. What is the smallest number with remainder 7 when you divide by 8, and remainder 6 when you divide by 7?

The numbers that have remainder 7 when divided by 8 are one less than multiples of 8:

8-1, 16-1, 24-1, and so on: 7, 15, 23, 31, 39, 47, 55, 63, …

The numbers that have remainder 6 when divided by 7 are one less than multiples of 7:

7-1, 14-1, 21-1, and so on: 6, 13, 20, 27, 34, 41, 48, 55, 62, …

The smallest match is 55. Another solution method is to note that 7 and 8 have no factors in common, so the smallest solution will be 7×8 – 1 = 56 – 1 = 55.

3. Four hamsters have 27 kibbles. They will divide the kibbles evenly among themselves and give the remainder to their friends, the mice. How many kibbles does each hamster get? How many kibbles go to the mice?

27 divided by 4 is 6 with remainder 3. The hamsters each get 6 kibbles, and the mice have to share 3 kibbles.

4. If you divide 80857 by 101, what is the quotient and what is the remainder?

The number 808 is a multiple of 101: 808 = 8 x 101

80800 is a multiple of 100.

Therefore 80800 is a multiple of 101: 80800 = 8 x 101 x 100

Therefore the quotient is 8 x 100 = 800, and the remainder is 80857 – 80800 = 57.

5. The remainders of 7 in multiples of 8 form a sequence. What is this sequence? (Look at the sequence of multiples of 8. Divide each multiple by 7. What is the sequence of remainders?)

Multiples of 8: 8, 16, 24, 32, 40, 48, 56, 64, 72,

Dividing by 7 and taking remainder: 8 divided by 7 = 1 remainder 1.

16 divided by 7 = 2 remainder 2.

And so on: the remainders form the sequence 1, 2, 3, 4, 5, 6, 0, 1, 2, 3, 4, 5, 6, 0, …

**Week 3: Remainders – Day 4**

1. What is the smallest number that has a remainder of 1 when you divide it by both three and four?

A good number to try would be (3 x 4) + 1, which is 13.

Of all the smaller numbers that have a remainder of 1 when divided by 4, 5 has remainder 2 when divided by 3, and 9 has remainder 0 when divided by 3. So 13 is the smallest.

2. What is the smallest number that has a remainder of 2 when you divide it by 3, and a remainder of 4 when you divide it by 5?

Numbers that have a remainder of 4 when divided by 5 are 4, 9, 14, 19, 24, 29, and so on.

The smallest such number that has a remainder of 2 when divided by 3 is 14.

2. When Mr. Buntle arranges his class into groups of four, there are three children left over. When Mr. Buntle arranges his class into groups of five, there are two children left over. All classes in Mr. Buntle’s school have at least twenty children. How many children are in Mr. Buntle’s class?

The first possible number is 22 (which is 5×4 + 2). But when 22 is divided by 4, the remainder is 2. The next possibility is 27 (which is 5×5 + 2). When 27 is divided by 4, the remainder is 3. So the number of children is 27.

3. A candy machine makes 8 candies in each batch. The candies are sold in boxes of 6 candies. What is the minimum number of batches of candies that can be made so that they can be put into boxes with no candies left over?

Three batches would make four boxes with none left over.

4. One number between 40 and 50 has a remainder of 7 when it is divided by 13. What is the number?

The multiples of 13 are 13, 26, 39, and so on. The nearest multiple to 40 is 39. 39 + 7 is 46. The number is 46.

5. One number between 30 and 40 has a remainder of 4 when it is divided by 7 and a remainder of 11 when it is divided by 14.

The nearest multiple of 14 to 30 is 28. 28 + 11 is 39. When 39 is divided by 7, the answer is 5 with remainder 4. The number is 39.

**Week 3: Remainders – Day 3**

**For each division, write an equation that describes the number as a multiple plus a remainder.**

1. Divide 12 by 5.

12 = 2×5 + 2

2. Divide 20 by 7.

20 = 2×7 + 6

3. Divide 46 by 4.

46 = 11×4 + 2

4. Divide 28 by 13.

28 = 2×13 + 2

5. Divide 59 by 11.

59 = 5×11 + 4

**Week 3: Remainders – Day 2**

Find the quotient and the remainder.

1. Divide 37 by 35.

37 = 35 + 2

quotient = 1, remainder = 2

2. Divide 89 by 22.

nearest multiple of 22 lower than 89 is 88. 88 is 4×22.

quotient = 4, remainder = 1.

3. Divide 427 by 10.

quotient = 42, remainder = 7.

4. Divide 427 by 100.

quotient = 4 (there are four hundreds), remainder = 27.

5. Divide 427 by 1000.

quotient = 0 (zero), remainder = 427.

**Week 3: Remainders – Day 1**

For each problem, list the quotient and the remainder.

1. Divide 6 by 4.

6 = 4 + 2.

The quotient is 1, and the remainder is 2.

2. Divide 11 by 2.

11 = 5×2 + 1.

The quotient is 5. The remainder is 1.

3. Divide 10 by 5.

10 = 5×2.

The quotient is 2. The remainder is zero.

4. Divide 11 by 10.

11 = 10 + 1.

The quotient is 1. The remainder is 1.

5. Divide 23 by 3.

23 = 21 + 2 = 7×3 + 2.

The quotient is 7. The remainder is 2.