Number Theory

Consider the following C program : #include int gate (int n) { int d, t, newnum, turn; newnum = turn = 0; t=1; while (n>= t) t*= 10; t/=10; while (t>0) { d=n/t; n=n%t; t/= 10; if (turn) newnum = 10*newnum + d; turn = (tu...

Let $p_1$ and $p_2$ denote two arbitrary prime numbers. Which one of the following statements is correct for all values of $p_1$ and $p_2$ ?

The number of coins of ₹1, ₹5, and ₹10 denominations that a person has are in the ratio 5:3:13. Of the total amount, the percentage of money in ₹5 coins is

If $${\left( {x - {1 \over 2}} \right)^2} - {\left( {x - {3 \over 2}} \right)^2} = x + 2$$, then the value of x is :

If P = 3, R = 27, T = 243, then Q + S = ______.

The value of 3 51 mod 5 is_________.

What is the missing number in the following sequence? $$$2,\,12,\,60,\,240,\,720,\,1440,\,\_\_\_,\,0$$$

What would be the smallest natural number which when divided either by $$20$$ or by $$42$$ or by $$76$$ leaves a remainder of $$7$$ in each case?

What would be the smallest natural number which when divided either by $$20$$ or by $$42$$ or by $$76$$ leaves a remainder of $$7$$ in each case?

If $$pqr \ne 0$$ and $${p^{ - x}} = {1 \over q},{q^{ - y}} = {1 \over r},\,{r^{ - z}} = {1 \over p},$$ what is the value of the product $$𝑥𝑦𝑧$$?

X is a 30 digit number starting with the digit 4 followed by the digit 7. Then the number X³ will have

Consider a quadratic equation $${x^2} - 13x + 36 = 0$$ with coefficients in a base $$b.$$ The solutions of this equation in the same base $$b$$ are $$x=5$$ and $$x=6$$. Then $$b=$$ ______.

Find the smallest number $y$ such that $y \times 162$ is a perfect cube.

If $$f\left( x \right) = 2{x^7} + 3x - 5,$$ which of the following is a factor of $$f(x)$$?

The value of the expression $${13^{99}}$$ ($$mod$$ $$17$$), in the range $$0$$ to $$16,$$ is ______________ .

In a quadratic function, the value of the product of the roots $$\left( {\alpha ,\beta } \right)$$ is $$4.$$ Find the value of $$${{{\alpha ^n} + {\beta ^n}} \over {{\alpha ^{ - n}} + {\beta ^{ - n}}}}$$$

The number of divisors of $$2100$$ is ___________.

There are 5 bags labeled 1 to 5. All the coins in given bag have the same weight. Some bags have coins of weight 10 gm, other have coins of weight 11 gm. $${\rm I}$$ pick 1, 2, 4, 8, 16 coins respectively from bags 1 to...

The number of distinct positive integral factors of 2014 is _______ .

The exponent of $$11$$ in the prime factorization of $$300!$$ is

The minimum positive integer p such that (3 p modulo 17) = 1 is

Let $$n = {p^2}q,$$ where $$p$$ and $$q$$ are distinct prime numbers. How many numbers $$m$$ satisfy $$1 \le m \le n$$ and $$gcd\left( {m.n} \right) = 1?$$ Note that $$gcd(m,n)$$ is the greatest common divisor of $$m$$ a...