The odd number in a group

Learning outcome

Find the number rule shared by three values, then mark the number that breaks it.

Concepts and reasons

Test a simple rule before a clever one. The rules in this lesson are: perfect square, perfect cube, and odd square. A number is a perfect square when it is an integer times itself. A perfect cube is an integer multiplied by itself twice.

Check the three that fit before naming the odd one. 8 = 2 cubed, 27 = 3 cubed and 64 = 4 cubed. The number 100 is 10 squared, not a cube, so 100 is outside that group.

Worked examples

Example 1. 8, 27, 64, 100. The first three are cubes. 100 is not a cube. The odd number is 100.

Example 2. 9, 25, 49, 50. The first three are squares of odd integers: 3, 5 and 7. The number 50 is not a square. The odd number is 50.

Common traps

Do not call a number odd only because it is the largest. Size is a rule only when the other three share a size pattern and the fourth does not. Here the shared pattern is square or cube.

Practice questions

1. Which number is outside 16, 36, 64, 81 if the class is even squares?
2. Is 1 both a square and a cube? What does that mean for a group?

Answers and explanations

1. 16 = 4 squared, 36 = 6 squared and 64 = 8 squared are even squares. 81 = 9 squared is an odd square, so 81 is outside that class.
2. Yes. 1 = 1 squared = 1 cubed. It can sit in either class, so it is a poor odd-one-out unless the other numbers force the class.

Analogy

8, 27, 64, 100.

Three are cubes. 100 is not. The odd number is 100.

Quick reference

• 8, 27, 64 and 125 are the cubes of 2, 3, 4 and 5.
• 9, 25, 49 and 81 are the squares of 3, 5, 7 and 9.
• Name the rule from the three that fit.
• Largest is not a rule by itself.