IIT JEE , AIEEE Forum - Mathematics , Algebra

neeraj_agarwal_1990

number of ordered triplets (p,q,r) where 1<= p,q,r<= 10 ,such that ${2}^{p}+{3}^{q}+{5}^{r}$ is a multiple of 4; is? (p,q,r (- N)
ans-500

Conjurer

Write 3^q = (4-1)^q and 5^r = (4+1)^r

Equation reduces to 2^p + 4m + (-1)^q + 4n + 1 = 4m (Applying binomial)

Obviously if p>=2 2^p is a multiple of 4

So taking p>=2, we get

4l + (-1)^q +1 = 4m

This implies for all values of p>=2 and r>=1 , we get the equation to be a multiple of 4 if q is odd.

So total no. of values this way is 9*5*10 = 450

Now if p=1, we get the equation to be a multiple of 4 iff q=even.

So no. of values in this case is 1*5*10 = 50

So total values = 500

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