Question

$N = \left[\frac{2023}{4202}\right] + \left[\frac{2023 \times 2}{4202}\right] + \dots + \left[\frac{2023 \times 4201}{4202}\right], \text{where [.] represents greatest integer function, then sum of digits of N is divisible by}$

Answer 1

21

Answer 2

The problem asks us to find the sum of digits of the number $N$, where $N$ is defined as: $N = \left[\frac{2023}{4202}\right] + \left[\frac{2023 \times 2}{4202}\right] + \dots + \left[\frac{2023 \times 4201}{4202}\right]$ Here, $[.]$ denotes the greatest integer function.

Let $a = 2023$ and $b = 4202$. The expression for $N$ can be written as: $N = \sum_{k=1}^{b-1} \left[\frac{ak}{b}\right]$

This sum has a well-known property. If $a$ and $b$ are coprime integers (i.e., $\gcd(a,b)=1$), then the sum is given by the formula: $\sum_{k=1}^{b-1} \left[\frac{ak}{b}\right] = \frac{(a-1)(b-1)}{2}$

First, let's check if $a=2023$ and $b=4202$ are coprime. Factorize $a$: $2023 = 7 \times 289 = 7 \times 17^2$ Factorize $b$: $4202 = 2 \times 2101$ To check if 2101 is prime, we can test divisibility by small prime numbers. $2101 \div 7 \approx 300.14$ $2101 \div 11 = 191$ So, $4202 = 2 \times 11 \times 191$. The prime factors of $a$ are $\{7, 17\}$. The prime factors of $b$ are $\{2, 11, 191\}$. Since there are no common prime factors, $\gcd(2023, 4202) = 1$. Thus, the formula is applicable.

Now, substitute the values of $a$ and $b$ into the formula for $N$: $N = \frac{(2023-1)(4202-1)}{2}$ $N = \frac{2022 \times 4201}{2}$ $N = 1011 \times 4201$

Next, calculate the value of $N$: $N = 1011 \times 4201$ $N = (1000 + 11) \times 4201$ $N = 1000 \times 4201 + 11 \times 4201$ $N = 4201000 + (10 \times 4201 + 1 \times 4201)$ $N = 4201000 + 42010 + 4201$ $N = 4201000 + 46211$ $N = 4247211$

Finally, find the sum of the digits of $N$: Sum of digits $= 4 + 2 + 4 + 7 + 2 + 1 + 1$ Sum of digits $= 21$

The question asks what the sum of digits of N is divisible by. The sum of digits is 21. The divisors of 21 are 1, 3, 7, and 21. Depending on the options provided in a multiple-choice question, any of these could be the correct answer. For example, if the options were 2, 3, 5, 9, then 3 would be the correct answer. If the options were 4, 6, 7, 8, then 7 would be the correct answer.

Given that no options are provided, we state the sum of digits and its properties. The sum of digits of N is 21, which is divisible by 3 and 7.