Arun Sharma Solutions for Exercise 3: Level of Difficulty

Suppose the sum of $\mathrm{n}$ consecutive integers is $x+\left(x+1\right)+\left(x+2\right)+\left(x+3\right)+.....+\left(x+(n-1\right))=1000$, then which of the following cannot be true about the number of terms $n$

$N=202\times 20002\times 200000002\times 20000000000000002\times 20000000...2\left(31 \mathrm{zeroes}\right)$. The sum of digits in this multiplication will be

Twenty five sets of problems on Data Interpretation one each for the DI sections of $25$ CATALYST tests were prepared by the AMS research team. The DI section of each CATALYST contained $50$ questions of which exactly $35$ questions were unique, i.e. they had not been used in the DI section of any of the other $24$ CATALYSTS. What could be the maximum possible number of questions prepared for the DI sections of all the $25$ CATALYSTS put together?

In the above question, what could be the minimum possible number of questions prepared?

How many times would $1$ be used while writing all the natural numbers from $8$ to $127$ in the Binary number system?

What is the maximum number of elements that one can pick from the set of natural numbers from $1$ to $20$ such that the product of no two of them results in a perfect square or perfect cube?

$abcdefghij$ is a ten-digit number with distinct digits such that $a > b > c, d > e > f, g > h > i > j$. $a, b, c$ are consecutive even digits and $g, h, i, j$ are consecutive odd digits.If $d + e + f = 9$, then what is the value of $\left[\frac{a \times b \times c \times d}{i}\right]$ where [ ] denotes greatest integer function?

$N = abc$ is a three digit number, the sum of whose digits is $\frac{1}{7}$th of the product of its digits. Then how many possible sets of $(a, b, c)$ are possible?

 ${\left(3132!\right)}_{10} = {\left(x\right)}_{34}$, then what will be the number of consecutive zeroes at the end of '$x$' ?