Question

Let $E=[\frac{1}{3}+\frac{1}{50}]+[\frac{1}{3}+\frac{2}{50}]+...+$ up to $50$ terms, then which of the following is true?
Here [] denotes the greatest integer function.

• A. $E$ is divisible by exactly $2$ primes
• B. $E$ is prime
• C. $E\le 30$
• D. $E\ge 35$

Answer 1

Answer: (B), (C)

$E$ is prime
$E\le 30$

For $1\le x\le 33,\frac{1}{3}<\frac{1}{3}+\frac{x}{50}<1$

Since, $\frac{1}{3}+\frac{1}{50}=\mathrm{0.3333..}+0.02=\mathrm{0.35333..}$ and $0.3333<0.3533<1$

Also, $\frac{1}{3}+\frac{33}{50}=\mathrm{0.3333..}+0.6666=\mathrm{0.99999..}$ and $0.3333<0.9999<1$

Therefore, $\frac{1}{3}+\frac{x}{50}=0$ for $1\le x\le 33$

For $34\le x\le 50$, $1<\frac{1}{3}+\frac{x}{50}<\frac{4}{3}$

Therefore, $\frac{1}{3}+\frac{x}{50}=1$ for $34\le x\le 50$

Thus $E=0+0+......+0+1+1+...1=17$

Hence, $E\le 30$ and $E$ is prime.