Find the remaining of \(9^{10^{11}}-5^{9^{10}}\) when divide for 13.

Find the remaining of \(3^{1998}+5^{1998}\) when divide for 13.

Find the remaining of \(1992^{1993}+1994^{1995}\) when divide for 7.

Find the remaining of \(2^{1994}\)  and \(3^{1993}\) when divide for 7.

Find the remaining of \(22^{22}+55^{55}\) when divide for 7.

Prove that: \(x^2-x^9-x^{1945}⋮x^2-x+1\)

Prove that: \(x^{50}+x^{10}+1⋮x^{20}+x^{10}+1\)

Prove that: \(x^{99}+x^{88}+x^{77}+...+x^{11}+1⋮x^9+x^8+x^7+...+x+1\)

Prove that: \(\left(x+y\right)^6+\left(x-y\right)^6⋮x^2+y^2\)

Given \(A=2013^{2014}+2014^{2013}\). Find the last digit number of A.

In a golf long-drive competition, Jason Zuback hits his first five drives 394, 401, 387, 414 and 421 yards, respectively. How long must he hit his sixth drive to ensure that the mean of his six drives is at least 400 yards?

What is the mean number of days per month among all months in the year 2018? Express your answer as a decimal to the nearest tenth.

For each of the first eight prime numbers, Brian Edwards writes down all the number’s positive factors. What is the sum of all the numbers Brian writes down?

In a new version of Scrabbles, a sequence of letters is considered a word if the first and last letters are consonants and every letter in between is a vowel. In this game, how many four-letter words can be formed using each of the letters M, A, T, H, R, U, L, E and S no more than once?

The third term of a geometric sequence of integers is 45. The seventh term of the sequence is 3645. What is the least possible sum of the first five terms of the sequence?

The Richter scale is used to describe the strength of an earthquake. An increase of 1 point on the Richter scale represents a tenfold increase in the strength of an earthquake. How many times stronger is an earthquake rated 7.5 on the Richter scale compared to an earthquake rated 5? Express your answer in simplest radical form. 

The decibel is a unit used to describe the loudness of a sound. For every 20-decibel increase, a sound gets 10 times as loud. Normal conversation is about 60 decibels, and a loud rock concert is about 120 decibels. How many times as loud is a rock concert compared to normal conversation?

Zeus threw, on average, 12 lightning bolts per day in the month of March. During the first week in April, he averaged 15 lightning bolts per day. How many lightning bolts does Zeus need to throw per day on average for the rest of April to maintain a 12-bolt-per-day average over March and April? Express your answer to the nearest integer.

In the arithmetic sequence 1, 3, 5, 7, 9, 11, …, how many terms appear after the term 315 but before the term 639?

A point D is placed on the segment with endpoints (0, 8) and (8, 0), and a point E is placed on the segment with endpoints (−3, 0) and (0, −2). What is the shortest possible distance between D and E? Express your answer in simplest radical form.