If two integers p and q are in the ratio 4: 5 and their least common m...
Step 1: Question statement and Inferences
We are given that positive integers p and q are in the ratio 4: 5. Also, the least common multiple of p and q is given as 800. We have to find the greatest common divisor of the given numbers.
Since p and q are in the ratio 4: 5, we can write them as:
p = 4x
q = 5x
Step 2: Finding required values
Given that:
p = 22 * x
q = 51 * x
Least common multiple = 22 * 51 * x (Since the prime numbers that make x will be common in both the numbers)
Greatest common divisor = x (Since there is no other common factor in the two numbers)
As we know, the product of least common multiple and greatest common divisor of two numbers is equal to the product of the numbers themselves, we can say:
4x * 5x = 800 * x
20 * x2 = 800 * x
So, x = 40
Step 3: Calculating the final answer
So, the greatest common divisor of p and q will be 40.
Answer: Option (B)
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If two integers p and q are in the ratio 4: 5 and their least common m...
Given:
- Two integers p and q are in the ratio 4:5.
- The least common multiple (LCM) of p and q is 800.
To find:
The greatest common divisor (GCD) of p and q.
Solution:
Step 1: Express the ratio of p and q as 4x and 5x, where x is a positive integer.
- Since the ratio of p and q is given as 4:5, we can express this as 4x:5x, where x is a positive integer.
Step 2: Find the LCM of p and q using the given ratio.
- The LCM of two numbers can be found by multiplying the numbers together and dividing by their GCD.
- LCM(p, q) = (4x * 5x) / GCD(p, q) = 20x^2 / GCD(p, q)
Step 3: Determine the LCM of p and q using the given information.
- LCM(p, q) = 800
Substituting this into the equation from Step 2:
800 = 20x^2 / GCD(p, q)
Step 4: Simplify the equation and find the value of x.
- Multiply both sides of the equation by GCD(p, q):
800 * GCD(p, q) = 20x^2
- Divide both sides of the equation by 20:
40 * GCD(p, q) = x^2
- Since x is a positive integer, x^2 must be a perfect square.
- Therefore, x must be a multiple of 40.
Step 5: Find the GCD of p and q.
- Since the ratio of p and q is 4x:5x, and x is a multiple of 40, the GCD of p and q must also be a multiple of 40.
- Therefore, the GCD of p and q is 40.
Answer:
The greatest common divisor (GCD) of p and q is 40.
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