For an integer \(n\), each of the following equations has at least one solution except which one?

\(a\) and \(b\) are given as distinct integers. If \(a^b = b^a\), how many ordered pairs \((a, b)\) are possible?

\(x\) and \(y\) are nonnegative integers satisfying \(2^x + 2^y = x^2 + y^2\). What is the greatest possible value of \(|x - y|\)?

find the units digit of \(2^{(1!+2!+3!+\cdots+10!)} + 3^{(1!+2!+3!+\cdots+10!)} + 4^{(1!+2!+3!+\cdots+10!)} + 5^{(1!+2!+3!+\cdots+10!)} + \cdots + 9^{(1!+2!+3!+\cdots+10!)}\).

if \((5.55)^x = (0.555)^y = 1000\), then what is the value of \(\frac{1}{x}-\frac{1}{y}\)?

If \((10)^9 + 2(11)^1(10)^8 + 3(11)^2(10)^7 + \cdots + 10(11)^9 = k(10)^9\), then \(k\) is equal to

if \(2^{2x+4}-17\times 2^{x+1}=-4\), then what is the sum of all possible values of \(x\)?

Let \(n\) and \(x\) be negative numbers. Also, \(n\) is an odd integer.

Which of the following statements could be true?

I. \(x^n < \frac{1}{x^n} < nx\) II. \(\frac{1}{x^n} < nx < x^n\) III. \(\frac{1}{x^n} < x^n < nx\)

\(Q\) represents the number of even integers \(x\) such that \(2^{20}
Which of the following must be true?

I. \(Q\) is even. II. \(Q\) is divisible by \(31\). III. \(Q\) is divisible by \(32\).

If \(n\) is a positive integer less than \(10\), \(k\) is a positive integer greater than \(10\), and \(n^{k+4}+n^{k+3}+n^{k+2}+n^{k+1}\) is divisible by \(10\), then \(n\) could be equal to which of the following?

I. \(3\) II. \(7\) III. \(8\)

Let \(a,b,c,d,e,f,g,h,i,j\) be distinct positive integers. What is the units digit of the lowest possible value of \(a^b \times c^d \times e^f \times g^h \times i^j\)?

find the number of integer values of \(x\) such that \(0 \le 2^x \le 200\), and \(2^x+2\) is divisible by either \(3\) or \(4\).

what is the remainder when \(32^{32^{32}}\) is divided by \(9\)?

\(x\) and \(y\) are integers such that \(x^y=256\). How many different values of \(y^x\) are possible?

find the number of trailing zeros in the product \((1^1)\times(5^5)\times(10^{10})\times(15^{15})\times(20^{20})\times(25^{25})\times\cdots\times(50^{50})\).

if \(2^{2x+4}-17\times 2^{x+1}=-4\), then which of the following is true?

If \(3^{3+6+9+\cdots+3x}=(0.\overline{037})^{-66}\), then what is the value of \(x\)?

If \(4^{6+12+18+24+\cdots+6x}=(0.0625)^{-84}\), then what is the value of \(x\)?

if \(x=1^1\times 2^2\times 3^3\times 4^4\times \cdots \times 98^{98}\times 99^{99}\times 100^{100}\), how many trailing zeros does \(x\) have?

If \(P=2^{x^2+10x+72}\) and \(Q=2^{x^2+16x+54}\), where \(x\) is a positive integer, and the least common multiple of \(P\) and \(Q\) is \(P\), then what is the number of values that \(x\) can take?

let \(N=10^{40}+2^{40}\). If \(2^k\) divides \(N\), but \(2^{k+1}\) does not divide \(N\), where \(k\) is a positive integer, then what is the value of \(k-2\)?

if \(0.7^{2x^2-3x+4}<0.343\), then which of the following must be true?

what is the nearest value of \(\frac{(0.888888)^{27}\times(0.333333)^6}{(0.592592)^{20}\times 0.444444}\)?

what is the greatest integer less than or equal to \(\frac{3^{100}+2^{100}}{3^{96}+2^{96}}\)?

If \(x=3+3^{\frac{1}{3}}+3^{\frac{5}{3}}\), then find the value of \(x^3-9x^2\).