p is a prime, n a positive integer and n+p =2000. LCM of n and p is 21879. Then ______________ and                .

@allarmarju i understand my mistake

so first we have to prove n and p are co primes

 

let n and p not be co primes

n=xp           where x is an integer

xp+p=2000              and xp=21879

x(p+1)=2000           and xp=21879

from this we donot reach to any solution

thus n and p are co primes

LCM * HCF=product of two numbers

HCF of co primes is always 1

np=21879

n+p=2000

n=2000-p

(2000-p)p=21879

solving equation we get

p=1989 and 11

but 1989 is not a prime as it is divisible by 3

11 is

so other integer is 21879 / 11=1989

so the prime no in the ans is 11 and integer is 1989

p=11     n=1989

PLS CORRECT IF WRONG

Here there are two cases,If the positive integer n  is a multiple of the prime p,I mean n=kp for some integer k then their lcm is kp itself.

So,(k+1)p=2000 and kp=21879

From this,we don't get any solution.

Therefore,nkp.So,n and p are co-primes,their lcm becomes np.

Hence,np=21879 and n+p=2000.

So,(n-p)²=(n+p)²-4npIn-pI=1978

Thus,p=11 or 1989,but p can't be 1989.So,the only solution is p=11 and n=1989.

Oh..I didn't notice the above post.But rahul,hcf of a prime and a positive integer is not always 1.If the positive integer is a multiple of prime,then their hcf is the prime itself and not 1.

edited