Modular Arithmetic: all problems

What is the units digit of $13^{2012}$?

What is the units digit of $19^{19} + 99^{99}$?

What is the tens digit of $7^{2011}$?

What is the smallest number greater than 2 that leaves a remainder of 2 when divided by 3, 4, 5, or 6?

The number $N$ is a two-digit number.

• When $N$ is divided by $9$, the remainder is $1$.
• When $N$ is divided by $10$, the remainder is $3$.

What is the remainder when $N$ is divided by $11$?

The product of the two 99-digit numbers $303{,}030{,}303{,}\ldots{,}030{,}303$ and $505{,}050{,}505{,}\ldots{,}050{,}505$ has thousands digit $A$ and units digit $B$. What is the sum of $A$ and $B$?

How many positive three-digit integers have a remainder of 2 when divided by 6, a remainder of 5 when divided by 9, and a remainder of 7 when divided by 11?

What is the units digit of $13^{17} + 17^{13}$?

Find the units digit of $7^{109} - 3^{336}$

Find the smallest positive number greater than 1 that leaves a remainder of 1 when divided by 2, by 3, by 4, by 5, and by 6.

How many positive integers can fill the blank in the sentence below?

“One positive integer is _____ more than twice another, and the sum of the two numbers is 28.”

The Fibonacci sequence $1, 1, 2, 3, 5, 8, 13, 21, \ldots$ starts with two 1s, and each term afterwards is the sum of its two predecessors. Which one of the ten digits is the last to appear in the units position of a number in the Fibonacci sequence?

(A) 0 · (B) 4 · (C) 6 · (D) 7 · (E) 9

If $n$ is an even positive integer, the double factorial notation $n!!$ represents the product of all the even integers from 2 to $n$. For example, $8!! = 2 \cdot 4 \cdot 6 \cdot 8$. What is the units digit of the following sum?

Find the remainder when $1^2 + 2^2 + 3^2 + \cdots + 2016^2$ is divided by 17.

Let $N = 100^2 + 99^2 - 98^2 - 97^2 + 96^2 + \cdots + 4^2 + 3^2 - 2^2 - 1^2$, where the additions and subtractions alternate in pairs. Find the remainder when $N$ is divided by 1000.

In the Fibonacci sequence, how many of the first 100 terms in the sequence leave a remainder of less than 7 when divided by 8?