Number of ordered triplets (p, q, r) where p, q, r ∈ N lying in [1,100], such that (2p+3q+5r) is divisible by 4 is λ ⋅ 105 where λ is equal to

Question

Number of ordered triplets (p, q, r) where p, q, r ∈ N lying in [1,100], such that (2p+3q+5r) is divisible by 4 is λ ⋅ 105 where λ is equal to (A) 50 (B) (1/2) (C) 5 (D) (1/20)

Tardigrade Admin · Accepted Answer

2 p +(4-1) q +(4+1) r ∴(2 p +(-1) q +(1) r ) text must be divisible by 4 p =1 text and q = text even ⇒ 1 × 50 × 100 p >1 text and q = text odd ⇒ 99 × 50 × 100 ∴ N =100 × 50 × 100=5 × 105

Q. Number of ordered triplets $(p, q, r)$ where p, q, r $\in N$ lying in $[1, 100]$ , such that $(2^{p} + 3^{q} + 5^{r})$ is divisible by 4 is $λ \cdot 1 0^{5}$ where $λ$ is equal to

50

$\frac{1}{2}$

5

$\frac{1}{20}$

$2^{p} + (4 - 1)^{q} + (4 + 1)^{r}$
$∴ (2^{p} + (- 1)^{q} + (1)^{r}) must be divisible by 4$
$p = 1 and q = even \Rightarrow 1 \times 50 \times 100$
$p > 1 and q = odd \Rightarrow 99 \times 50 \times 100$
$∴ N = 100 \times 50 \times 100 = 5 \times 1 0^{5}$

Q. Number of ordered triplets (p,q,r) where p, q, r ∈N lying in [1,100], such that (2p+3q+5r) is divisible by 4 is λ⋅105 where λ is equal to

50

21​

5

201​

2p+(4−1)q+(4+1)r ∴(2p+(−1)q+(1)r) must be divisible by 4 p=1 and q= even ⇒1×50×100 p>1 and q= odd ⇒99×50×100 ∴N=100×50×100=5×105

Q. Number of ordered triplets $(p, q, r)$ where p, q, r $\in N$ lying in $[1, 100]$ , such that $(2^{p} + 3^{q} + 5^{r})$ is divisible by 4 is $λ \cdot 1 0^{5}$ where $λ$ is equal to

$\frac{1}{2}$

$\frac{1}{20}$

$2^{p} + (4 - 1)^{q} + (4 + 1)^{r}$
$∴ (2^{p} + (- 1)^{q} + (1)^{r}) must be divisible by 4$
$p = 1 and q = even \Rightarrow 1 \times 50 \times 100$
$p > 1 and q = odd \Rightarrow 99 \times 50 \times 100$
$∴ N = 100 \times 50 \times 100 = 5 \times 1 0^{5}$