What is the value of the expression \text{GCD}(2A + B, 2B + A) given the conditions about LCM?
Answer:
To solve for the greatest common divisor (GCD) of the expressions 2A + B and 2B + A, we use the information given about the least common multiple (LCM) of A, A+1, B, and B+1. We are given:
- \text{LCM}(A, A+1) = 420
- \text{LCM}(B, B+1) = 210
Step 1: Use LCM Properties
For any two consecutive integers, say n and n+1, they are always coprime, meaning \text{GCD}(n, n+1) = 1. Therefore, \text{LCM}(n, n+1) = n(n+1).
Thus, we have:
These are two separate equations to solve for positive integer values A and B.
Step 2: Solve Each Equation
First, consider A(A+1) = 420. We look for integers A such that this equation holds:
Factors of 420 are: 1, 2, 3, 4, 5, 6, …, 420.
Trying A = 20:
This works.
Now, consider B(B+1) = 210. We proceed similarly:
Factors of 210 are: 1, 2, 3, 5, 6, …, 210.
Trying B = 14:
This works.
So, the possible values are A = 20 and B = 14.
Step 3: Find GCD of Expressions
Now, compute \text{GCD}(2A + B, 2B + A).
Substitute A = 20 and B = 14:
- 2A + B = 2(20) + 14 = 54
- 2B + A = 2(14) + 20 = 48
Now, find \text{GCD}(54, 48):
- The prime factors of 54 are 2 \times 3^3
- The prime factors of 48 are 2^4 \times 3
The common factors are 2 and 3, where the minimum powers are 2^1 and 3^1, giving:
Final Answer:
The value of the expression \text{GCD}(2A + B, 2B + A) is 6.