Partners and the odd one out
Factors pair up — except in a square.
The locker puzzle
100 lockers, all closed. Person 1 opens every locker. Person 2 toggles every 2nd locker. Person 3 toggles every 3rd… up to Person 100.
A person toggles a locker when their number is a factor of the locker number. Locker 6 is toggled by persons 1, 2, 3 and 6. Locker 9 is toggled by persons 1, 3 and 9. (Toggle = open it if closed, close it if open.)
At the end, which locker stays open?
What this lesson covers
The idea
Factors of a number come in pairs of partner factors whose product is the number; only squares have an odd number of factors, because one of their factors is its own partner.
The locker puzzle
100 lockers, all closed. Person 1 opens every locker. Person 2 toggles every 2nd locker. Person 3 toggles every 3rd… up to Person 100.
A person toggles a locker when their number is a factor of the locker number. Locker 6 is toggled by persons 1, 2, 3 and 6. Locker 9 is toggled by persons 1, 3 and 9. (Toggle = open it if closed, close it if open.)
At the end, which locker stays open?
- Locker 6
- Locker 9
- Both
Pair up the factors
Tap a number. Its factors are shown in partner pairs whose product is the number. Look for the pairs that do not have two different partners.
Partners
Factors come in partner pairs with product equal to the number. Only squares have an odd number of factors, because one factor is its own partner.
A locker stays open only if it is toggled an odd number of times. So only locker numbers that are squares stay open: 1, 4, 9, 16, 25, …
- Number | Partner pairs | Factors
- 6 | 1 × 6, 2 × 3 | 4 (even)
- 9 | 1 × 9, 3 × 3 | 3 (odd)
- 36 | 1 × 36, 2 × 18, 3 × 12, 4 × 9, 6 × 6 | 9 (odd)
Notes
Factors come in partner pairs with product equal to the number. Only squares have an odd number of factors, because one factor is its own partner.
Check yourself
Which of these numbers has an odd number of factors?
How many factors does 36 have?
Answer: 9
1, 2, 3, 4, 6, 9, 12, 18, 36: nine factors, an odd number, because 36 is a square.
In the locker puzzle with 100 lockers, how many lockers are open at the end?
Answer: 10
The squares 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 — ten lockers.
- 20. 20 = 1×20, 2×10, 4×5. Three pairs, all with different partners: 6 factors, an even number.
- 25 — correct. Yes! 25 = 1 × 25 and 5 × 5. The factor 5 is its own partner, so there are 3 factors.
- 28. 28 = 1×28, 2×14, 4×7. Three pairs: 6 factors, an even number.
- 30. 30 = 1×30, 2×15, 3×10, 5×6. Four pairs: 8 factors, an even number.