What are the possible values for p and q?

Question

What are the possible values for p and q if the Greatest Common Divisor between these two is 14 and their Least common multiple is 210?

Answer 1

The greatest common divisor is a number that divides both p and q.  So we know p=14*x and q=14*y for some values of x and y.  

 

The least common multiple is the first multiple of both p and q.  Since we know 14 divides both numbers we know that

x*y = 210/14 = 15.

 

We can find the prime factors of 15 and use them to find possible values for p and q.  For example 15 = 1*15, so one possible combination is p=14 and q=210.  You can factor 15 and multiply to find all other possibilities.  

Answer 2

If we multiply two numbers together, we get a product.  A product is the same as a multiple.

If we divide two numbers, we get a factor of the number we are dividing.  A factor is the same as a divisor.

 

Knowing this, we can create two equations.

 

pq = 210                  eq1

 

p/q = 14                   eq2

 

 

We have a system of equations.  If we substitute eq2 into eq1 so the eq1 is in terms of eq1, we obtain

 

(14q)q = 210

 

14q² = 210

 

 

Solve for q from this equation.

 

q² = 15

 

 

Square-root both sides of the equation.

 

q = √15

 

 

If you know the value of q, then can you figure out the value of p?