Is the product of two consecutive integers always multiple of 2? Why? What about the product of three consecutive integers? Is it always a multiple of 6? Why or why not? What can you say about the product of 4 consecutive integers? What about the product of five consecutive integers?
Product of Two Consecutive Integers: Always a multiple of 2.
Among any two consecutive integers, one must be even.
∴ The product is divisible by 2.
Product of Three Consecutive Integers: Always a multiple of 6.
Among any three consecutive integers:
- At least one is even (divisible by 2)
- Exactly one is divisible by 3
∴ The product is divisible by = 6.
Product of Four Consecutive Integers: Always a multiple of 24.
Among any four consecutive integers:
- At least two are even (one divisible by 4)
- One is divisible by 3
∴ The product is divisible by × 3 = 24.
Product of Five Consecutive Integers: Always a multiple of 120.
Among any five consecutive integers:
- At least two are even (one divisible by 4)
- One is divisible by 3
- One is divisible by 5
∴ The product is divisible by × = 120.
Explanation
The chapter uses reasoning about consecutive integers. In any set of n consecutive integers, there's always at least one multiple of each number from 1 to n. This is because the remainders cycle through 0, 1, 2, ..., n-1 when dividing consecutive integers by n.
Solution Steps
Two consecutive: one even → product divisible by 2
Three consecutive: one even, one divisible by 3 → product divisible by 6
Four consecutive: two even (one divisible by 4), one divisible by 3 → product divisible by 24
Five consecutive: two even (one divisible by 4), one divisible by 3, one divisible by 5 → product divisible by 120