Recent questions tagged number-systems

Let $n$ and $m$ be two positive integers such that there are exactly $41$ integers greater than $8^{m}$ and less than $8^{n}$, which can be expressed as powers of $2$. Th...

The sum of the first two natural numbers, each having 15 factors (including 1 and the number itself), is

If $x$ is a positive real number such that $x^{8}+\left(\frac{1}{x}\right)^{8}=47$, then the value of $x^{9}+\left(\frac{1}{x}\right)^{9}$ is$34 \sqrt{5}$$40 \sqrt{5}$$36...

Let $n$ and $m$ be two positive integers such that there are exactly $41$ integers greater than $8^{m}$ and less than $8^{n}$, which can be expressed as powers of $2$. Th...

For any natural numbers $\mathrm{m}, \mathrm{n}$, and $\mathrm{k}$, such that $\mathrm{k}$ divides both $m+2 n$ and $3 m+4 n \mathrm{k}$ must be a common divisor of$m$ an...

A shop owner bought a total of $64$ shirts from a wholesale market that came in two sizes, small and large. The price of a small shirt was $\text{INR} \; 50$ less than th...

If $n$ is a positive integer such that $( \sqrt[7]{10}) ( \sqrt[7]{10})^{2} \dots ( \sqrt[7]{10})^{n} 999,$ then the smallest value of $n$ is

For a $4$-digit number, the sum of its digits in the thousands, hundreds and tens places is $14,$ the sum of its digits in the hundreds, tens and units places is $15,$ an...

How many three-digit numbers are greater than $100$ and increase by $198$ when the three digits are arranged in the reverse order?

How many of the integers $1,2, \dots, 120,$ are divisible by none of $2,5$ and $7 ?$$40$$42$$43$$41$

If $\textsf{x}$ and $\textsf{y}$ are non-negative integers such that $\textsf{x+9=z, y+1=z}$ and $\textsf{x+y<z+5},$ then the maximum possible value of $\textsf{2x+y}$ eq...

Among $100$ students, $x_{1}$ have birthdays in January, $x_{2}$ have birthdays in February, and so on. If $x_{0}= \text{max}\left ( x_{1},x_{2},\dots,x_{12} \right ),$ t...

By which one of the following should we multiply $152207$ so that the product is $11111111$?$53$$63$$73$$83$

$x$ is a whole number. If the only common factors of $x$ and $x2$ are $1$ and $x,$ then $x$ is ________.$1$a perfect squarean odd numbera prime number

The value of $[(10)^{150}\div (10)^{146}]$:$1000$$10000$$100000$$10^{6}$

How many pairs $(m,n)$ of positive integers satisfy the equation $m^{2}+105=n^{2}$ _______

In a six-digit number, the sixth, that is, the rightmost, digit is the sum of the first three digits, the fifth digit is the sum of first two digits, the third digit is e...

The smallest integer $n$ such that $n^{3} - 11n^{2} + 32n - 28 >0$ is

How many two-digit numbers, with a non-zero digit in the units place, are there which are more than thrice the number formed by interchanging the positions of its digits?...

Let $t_{1}, t_{2},\dots$ be a real numbers such that $t_{1}+t_{2}+\dots+t_{n}=2n^{2}+9n+13$, for every positive integers $n\geq2$.If $t_{k}=103$ , then $k$ equals

If $\text{N}$ and $x$ are positive integers such that $\text{N}^{\text{N}}=2^{160}$ and $\text{N}^{2} + 2^{\text{N}}$ is an integral multiple of $2^{x}$, then the largest...

If the sum of squares of two numbers is $97$, then which one of the following cannot be their product?$-32$$48$$64$$16$

The smallest integer $n$ for which $4^{n}>17^{19}$ holds, is closest to$33$$37$$39$$35$