Two even numbers make a multiple of 4
Join the leftover cells: 2 and 2 fill a row of four.
Adding two even numbers
6 + 10 = 16 is a multiple of 4. 4 + 8 = 12 is too. But 6 + 8 = 14 is not.
When is the sum of two even numbers a multiple of 4?
What this lesson covers
The idea
The sum of two even numbers is a multiple of 4 when both are multiples of 4 or both are not (their remainders 2 add up to 4); a multiple of 4 plus a non-multiple of 4 leaves remainder 2.
Adding two even numbers
6 + 10 = 16 is a multiple of 4. 4 + 8 = 12 is too. But 6 + 8 = 14 is not.
When is the sum of two even numbers a multiple of 4?
- Always
- Never
- Sometimes: it depends on the two numbers
Join two numbers
Each number is laid out in rows of 4. Use the buttons to change A and B (steps of 2, or 4 to keep the kind). See what the leftover cells do when A and B are joined.
Remainders 2 and 2 make 4
The sum of two even numbers is a multiple of 4 when both are multiples of 4 or both are not (their remainders 2 add up to 4); a multiple of 4 plus a non-multiple of 4 leaves remainder 2.
Both multiples of 4: 4p + 4q = 4(p + q) Neither is a multiple of 4: (4p + 2) + (4q + 2) = 4(p + q + 1) One of each: 4p + (4q + 2) = 4(p + q) + 2
Examples: 12 + 16 = 28 6 + 10 = 16 22 + 6 = 28
Notes
The sum of two even numbers is a multiple of 4 when both are multiples of 4 or both are not (their remainders 2 add up to 4); a multiple of 4 plus a non-multiple of 4 leaves remainder 2.
Check yourself
Which sum is a multiple of 4?
What remainder does 52 + 38 leave when divided by 4?
Answer: 2
52 = 4 × 13 is a multiple of 4. 38 = 4 × 9 + 2 leaves 2. The sum 90 = 4 × 22 + 2 leaves remainder 2.
(4p + 2) + (4q + 2) = 4(p + q + ?). What number replaces the ?
Answer: 1
(4p + 2) + (4q + 2) = 4p + 4q + 4 = 4(p + q + 1).
Two even numbers add up to a number that leaves remainder 2 when divided by 4. What do you know about the two numbers?
- 6 + 12. 6 leaves 2 and 12 leaves 0. One of each: 6 + 12 = 18 leaves remainder 2.
- 8 + 10. 8 leaves 0 and 10 leaves 2. One of each: 8 + 10 = 18 leaves remainder 2.
- 14 + 22 — correct. Yes! 14 and 22 both leave remainder 2. The leftovers 2 + 2 = 4 fill a row. 14 + 22 = 36 = 4 × 9.
- 14 + 16. 14 leaves 2 and 16 leaves 0. One of each: 14 + 16 = 30 leaves remainder 2.
- Both are multiples of 4. Then the sum would be a multiple of 4 and leave remainder 0.
- Neither is a multiple of 4. Then the leftovers 2 + 2 = 4 would fill a row and the sum would be a multiple of 4.
- Exactly one is a multiple of 4 — correct. Yes! Only then is a single leftover 2 left unmatched.