Find a few numbers that leave a remainder of 2 when divided by 3 and a remainder of 2 when divided by 4. Write an algebraic expression to describe all such numbers.
Given: Find numbers that leave remainder 2 when divided by both 3 and 4.
Analysis: Numbers ≡ 2 (mod 3): form 3m + 2 Numbers ≡ 2 (mod 4): form 4n + 2 LCM(3, 4) = 12
Common form: Numbers satisfying both conditions must be 2 more than a multiple of 12.
General expression: 12k + 2, where k is a non-negative integer.
Examples: k = 0 → 12(0) + 2 = 2 k = 1 → 12(1) + 2 = 14 k = 2 → 12(2) + 2 = 26 k = 3 → 12(3) + 2 = 38 k = 4 → 12(4) + 2 = 50
Answer: Numbers: 2, 14, 26, 38, 50, ... Algebraic expression: 12k + 2, where k is a non-negative integer.
Explanation
The chapter explains that numbers leaving remainder 2 when divided by 3 are of the form 3m + 2. Numbers leaving remainder 2 when divided by 4 are of the form 4n + 2. The LCM of 3 and 4 is 12. Numbers satisfying both conditions must be 2 more than a multiple of 12, giving the form 12k + 2.
Solution Steps
Numbers leaving remainder 2 when divided by 3 are of the form 3m + 2
Numbers leaving remainder 2 when divided by 4 are of the form 4n + 2
Find LCM of 3 and 4, which is 12
Numbers satisfying both conditions must be 2 more than a multiple of 12
Write the general form as 12k + 2, where k is a non-negative integer
Substitute k = 0, 1, 2, 3, 4 to get examples: 2, 14, 26, 38, 50