Number System · Basic Numeracy · UPSC Civil Service

A natural number $N$ is such that it can be expressed as $\mathbf{N}=\mathbf{p}+\mathbf{q}+\mathbf{r}$, where $\mathbf{p}, \mathbf{q}$ and $\mathbf{r}$ are distinct factors of $\mathbf{N}$. How many numbers below 50 have this property?

How many possible values of $(p+q+r)$ are there satisfying $\frac{1}{p}+\frac{1}{q}+\frac{1}{r}=1$, where $p, q$ and $r$ are natural numbers (not necessarily distinct)?

What is the unit digit in the multiplication of $1 \times 3 \times 5 \times 7 \times 9 \times \ldots \times 999$ ?

Consider the first 100 natural numbers. How many of them are not divisible by any one of $2,3,5,7$ and 9 ?

Let both $p$ and $k$ be prime numbers such that $\left(p^2+k\right)$ is also a prime number less than 30 . What is the number of possible values of $k$ ?

Let PQR be a 3-digit number, PPT be a 3-digit number and $\mathbf{P S}$ be a 2-digit number, where $\mathbf{P}, \mathbf{Q}, \mathbf{R}, \mathbf{S}, \mathbf{T}$ are distinct non-zero digits. Further, $\mathrm{PQR}-\mathrm{PS}=\mathrm{PPT}$. If $Q=3$ and $T<6$, then what is the number of possible values of $(\mathbf{R}, \mathbf{S})$ ?

What is the maximum value of $n$ such that $7 \times 343 \times 385 \times 1000 \times 2401 \times 77777$ is divisible by $35^{\mathrm{n}}$ ?

If $N^2=12345678987654321$, then how many digits does the number N have?

If $n$ is a natural number, then what is the number of distinct remainders of $\left(1^n+2^n\right)$ when divided by 4 ?

What is the 489 th digit in the number $123456789101112 \ldots$ ?

The 5-digit number PQRST (all distinct digits) is such that $\mathbf{T} \neq \mathbf{0} . \mathbf{P}$ is thrice $\mathbf{T} . \mathbf{S}$ is greater than $\mathbf{Q}$ by 4, while $Q$ is greater than $R$ by 3 . How many such 5-digit numbers are possible?

Consider the following statements:

I. There exists a natural number which when increased by $50 \%$ can have its number of factors unchanged.

II. There exists a natural number which when increased by $150 \%$ can have its number of factors unchanged.

Which of the statements given above is/are correct?

What is the remainder when $9^3+9^4+9^5+9^6+\ldots+9^{100}$ is divided by 6 ?

The difference between any two natural numbers is 10 . What can be said about the natural numbers which are divisible by 5 and lie between these two numbers?

$222^{333} + 333^{222}$ is divisible by which of the following numbers?

Consider the following statements in respect of the sum $S = x + y + z$, where $x, y$ and $z$ are distinct prime numbers each less than 10:

1. The unit digit of $S$ can be 0.

2. The unit digit of $S$ can be 9.

3. The unit digit of $S$ can be 5.

Which of the statements given above are correct?

Let $X$ be a two-digit number and $Y$ be another two-digit number formed by interchanging the digits of $X$. If $(X + Y)$ is the greatest two-digit number, then what is the number of possible values of $X$?

$32^5 + 2^{27}$ is divisible by

Consider the following in respect of prime number p and composite number c.

1. $\frac{\mathrm{p}+\mathrm{c}}{\mathrm{p}-\mathrm{c}}$ can be even.

2. 2p + c can be odd.

3. pc can be odd.

Which of the statements given above are correct?

Consider the following statements in respect of two natural numbers p and q such that p is a prime number and q is a composite number :

1. p × q can be an odd number.

2. q / p can be a prime number.

3. p + q can be a prime number.

Which of the above statements are correct?

The difference between a 2-digit number and the number obtained by interchanging the positions of the digits is 54.

Consider the following statements:

1. The sum of the two digits of the number can be determined only if the product of the two digits is known.

2. The difference between the two digits of the number can be determined.

Which of the above statements is/are correct?