Modular Arithmetic

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Show your work for each problem.

1.
What is $23 \bmod 5$ ?
1. a)2
2. b)4
3. c)3
4. d)18
2.
Find $45 \bmod 7$ .
3.
It is 8 o'clock. What does a 12-hour clock show 7 hours later?
1. a)1 o'clock
2. b)11 o'clock
3. c)15 o'clock
4. d)3 o'clock
4.
Find $-5 \bmod 12$ (a remainder from 0 to 11).
5.
Find $58 \bmod 9$ .
6.
The numbers $a$ and $b$ are both odd. What are $(a + b) \bmod 2$ and $(a \times b) \bmod 2$ ?
1. a)1 and 1
2. b)0 and 1
3. c)0 and 0
4. d)1 and 0
7.
Today is Tuesday. Which day of the week will it be in 60 days?
1. a)Saturday
2. b)Thursday
3. c)Monday
4. d)Sunday
8.
It is 10 o'clock. Which number does the hour hand of a 12-hour clock point to 250 hours later? (Answer with a number from 1 to 12.)
9.
Find $(47 + 86) \bmod 9$ by reducing first.
10.
Find $(34 \times 29) \bmod 5$ by reducing first.
11.
Which statement is true?
1. a) $17 \equiv 5 \pmod{8}$
2. b) $38 \equiv 14 \pmod{12}$
3. c) $25 \equiv 4 \pmod{6}$
4. d) $40 \equiv 2 \pmod{3}$
12.
Check that $83 \equiv 20 \pmod{9}$ , then find the remainder that both numbers leave when divided by 9.
13.
It is 3 o'clock. Which number did the hour hand of a 12-hour clock point to 40 hours ago?
14.
What is the last digit of $7^{50}$ ?
15.
What is the last digit of $2^{2026}$ ?
16.
Find the EAN-13 check digit for the 12 digits $590123412345$ . Digits in odd positions count once, digits in even positions count three times, and the check digit makes the total a multiple of 10.
17.
An ISBN-10 starts with the 9 digits 0-306-40615. Multiply these digits by $10, 9, 8, \ldots, 2$ and add. The check digit $x$ (weight 1) makes the total a multiple of 11. Find $x$ .
18.
What is $5^{100} \bmod 7$ ?