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If the least common multiplier of positive integers A and B is 120 and the ratio of A and B is 3:4, what is the largest common divisor of A and B?
- a)8
- b)9
- c)10
- d)12
- e)15
Correct answer is option 'C'. Can you explain this answer?
Most Upvoted Answer
If the least common multiplier of positive integers A and B is 120 and...
Given:
- A: B = 3: 4
- LCM of A and B = 120
To find:
The GCD of A and B
The GCD of A and B
Approach and Working Out:
- We can say that A = 3x and B = 4x
- This implies, the GCD of A and B = x, since 3 and 4 are co-primes.
We also know that, LCM (A, B) * GCD (A, B) = A * B
- 120 * x = 3x * 4x
- Therefore, x = 10
Hence, the correct answer is Option C.
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Community Answer
If the least common multiplier of positive integers A and B is 120 and...
To find the largest common divisor (LCD) of two positive integers A and B, we need to understand the relationship between the least common multiple (LCM) and the LCD.
The LCM of two numbers is the smallest number that is divisible by both numbers. In this case, the LCM of A and B is given as 120.
We are also given that the ratio of A to B is 3:4. Let's assume that A = 3x and B = 4x, where x is a positive integer.
To find the LCM of A and B, we can calculate it using the formula:
LCM(A, B) = (A * B) / GCD(A, B)
where GCD(A, B) represents the greatest common divisor of A and B.
Substituting the values of A and B:
LCM(3x, 4x) = (3x * 4x) / GCD(3x, 4x)
120 = 12x^2 / GCD(3x, 4x)
Now, we know that the LCM of A and B is 120. Hence, (3x * 4x) / GCD(3x, 4x) = 120.
From this equation, we can deduce that GCD(3x, 4x) = (3x * 4x) / 120 = 12x^2 / 120 = x^2 / 10.
Since GCD(3x, 4x) must be an integer, x^2 must be divisible by 10.
The possible values of x that satisfy this condition are x = 1, 2, or 3.
Now, let's calculate the values of A and B for each value of x:
For x = 1:
A = 3x = 3 * 1 = 3
B = 4x = 4 * 1 = 4
For x = 2:
A = 3x = 3 * 2 = 6
B = 4x = 4 * 2 = 8
For x = 3:
A = 3x = 3 * 3 = 9
B = 4x = 4 * 3 = 12
From these calculations, we can see that the largest common divisor of A and B is 3, which is the value of A when x = 1.
Therefore, the correct answer is option C) 10.
The LCM of two numbers is the smallest number that is divisible by both numbers. In this case, the LCM of A and B is given as 120.
We are also given that the ratio of A to B is 3:4. Let's assume that A = 3x and B = 4x, where x is a positive integer.
To find the LCM of A and B, we can calculate it using the formula:
LCM(A, B) = (A * B) / GCD(A, B)
where GCD(A, B) represents the greatest common divisor of A and B.
Substituting the values of A and B:
LCM(3x, 4x) = (3x * 4x) / GCD(3x, 4x)
120 = 12x^2 / GCD(3x, 4x)
Now, we know that the LCM of A and B is 120. Hence, (3x * 4x) / GCD(3x, 4x) = 120.
From this equation, we can deduce that GCD(3x, 4x) = (3x * 4x) / 120 = 12x^2 / 120 = x^2 / 10.
Since GCD(3x, 4x) must be an integer, x^2 must be divisible by 10.
The possible values of x that satisfy this condition are x = 1, 2, or 3.
Now, let's calculate the values of A and B for each value of x:
For x = 1:
A = 3x = 3 * 1 = 3
B = 4x = 4 * 1 = 4
For x = 2:
A = 3x = 3 * 2 = 6
B = 4x = 4 * 2 = 8
For x = 3:
A = 3x = 3 * 3 = 9
B = 4x = 4 * 3 = 12
From these calculations, we can see that the largest common divisor of A and B is 3, which is the value of A when x = 1.
Therefore, the correct answer is option C) 10.
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