If n+p=2000 and LCM of n and p is 21879. Then find n and p

Question

P is prime and n is a positive integer

Answer 1

P is prime and n is a positive integer

P is prime means factor of p is 1 and p

so we can write

n*p=21879 ------> n=21879/p

n+p=2000  ------>n=2000-p

by equating both n

2000 - p = 21879/p

 

2000p-p²=21879

 

p² -2000p=-21879

 

p² -2000p + 21879 =0

 

(p-11)(p-1989)=0

p=11 and p=1989

 

as 11 is prime no. 

p=11

n= 1989

Answer 2

To solve this problem, we can set up a system of equations.

 

Thus:

n+p = 2000

np = 21879 ----Since p is prime, the 2 numbers do not have common factors except 1. LCM = product of n and p.

 

Solving equation 1 for n yields --> n = 2000-p

Substitute this in the 2nd equation. Thus:

 

(2000-p)p = 21879

 

2000p - p² = 21879

 

p² - 2000p + 21879 = 0; Factoring yields (p - 11)(p - 1989) = 0

 

p - 11 = 0 and p - 1989 = 0

 

Therefore p = 11 and p =1989

 

Since 11 is prime, p must be 11.

n = 2000 - 11 = 1989; this is a positive integer.

 

The numbers are: n = 1989 and p = 11.