Whole division, remainder and power

What you will be able to do

After this lesson you can

• divide with a whole-number answer and a remainder,
• use the remainder to test whether a number is even or odd,
• take a number apart into digits, and
• use ** for squares, cubes and roots.

Sharing sweets

47 sweets are shared among 5 children. Each child gets 9 sweets, and 2 sweets are left over. We can say this in three different ways, with three different signs.

The three fit together in a rule that is always true:

quotient * divisor + remainder = the number, which here is 9 * 5 + 2 = 47.

The last line of the program checks it. This rule is worth remembering, because it explains what // and % mean.

The formula under the picture is always true: the children's sweets and the leftover sweets together make all the sweets.

Four uses of the remainder

The remainder looks like a small thing, but programmers use it all the time.

By using // 10 and % 10 again and again, you can take any number apart into its digits. This is how a program adds the digits of a number.

Powers and roots

The sign ** is the power.

Negative numbers and floor division

For positive numbers, // just cuts off the decimal part. For negative numbers, it goes down. -7 // 2 is -4, not -3, because -4 is the next whole number below -3.5. You will not meet this often, but it explains a surprise if you see one.

Common mistakes

• Using / when you want a whole number. 47 / 5 is 9.4. Use 47 // 5 for 9.
• Mixing up // and %. // tells how many, and % tells what is left.
• Reading % as percent. In Python, % is the remainder. For 20 percent of 50, write 50 * 20 / 100.
• Writing ^ for a power. In Python, ^ is something else. Use **.
• Forgetting that the remainder is always smaller than the divisor. n % 5 is always one of 0, 1, 2, 3 or 4.

Practice

Start with two quick questions. Each New numbers gives you a fresh one, so you can practise as long as you like.

Then the labs and the problems:

1. Warm-up. Open the lab Quotient and remainder. Find the quotient and the remainder of three divisions. The checker confirms each one.
2. Build. Open the problem The last digit, and then do the problem Even or odd?
3. Stretch. Open the lab Take a number apart. Take the number 472 apart into hundreds, tens and ones, using only // and %.

Remember these

• // gives the whole number of times. % gives the remainder. ** is the power.
• quotient * divisor + remainder = the number.
• n % 2 is 0 for an even number. n % 10 is the last digit. n // 10 removes it.
• A power of 0.5 is a square root.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. What are 17 // 5 and 17 % 5?

2. Which test shows that n is even?

3. How do you get the tens digit of 472?

4. What does 16 ** 0.5 show?