Download Homework on modular arithmetic

Math 202 Homework Set A
2.2.2007
1. Calculate the following sums and differences:
(a) (17 + 15) mod 20
(b) (3 + 5) mod 7
(c) (18 + 23 + 14 − 3) mod 25
(d) (16 − 23) mod 40
(e) (15 − 3) mod 27
(f) (8 + 5 − 26) mod 54
2. Find three integers which are congruent to 5 mod 12.
3. Find three negative integers which are congruent to 3 mod 5.
4. Reduce the following expressions.
(a) 80567 mod 5
(b) 805673 mod 100
(c) 78431 mod 2
(d) 176543 mod 4
(e) 257883 mod 50
(f) 703 mod 7
5. True or False?
(a) 3|56
(b) 3|96
(c) 0|0
(d) 0|4
(e) 4|0
(f) 17|(17 · 54)
(g) 13|(13 · 20 + 5)
(h) 78 ≡ 15 mod 5
(i) 223 ≡ 171 mod 2
(j) 58 ≡ −12 mod 10
(k) 16 ≡ −56 mod 9
6. Construct an addition table for Z7 .
7. Calculate the following:
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Math 202 Homework Set A
2.2.2007
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(a) (17 · 5) mod 23
(b) (4 · 2) mod 5
(c) (15 · 8 + 12) mod 19
(d) (5 · 7 + 3) mod 8
(e) (3 · 9 + 2 · 5) mod 10
8. Construct a multiplication table for Z7 . What do you observe about the multiplication table (what symmetry exists in the numbers? what does each row
look like? etc.)
9. Construct a multiplication table for Z10 . What do the rows corresponding to
multiplication by 1, 3, 7, and 9 have in common? Do the other rows have this
property? What do you observe in the other rows?
10. Construct a multiplication table for Z12 . Look at the numbers corresponding
to multiplication by 3. What do all the numbers in this row have in common?
Is there a pattern in this row? How long does the pattern take to repeat itself?
11. Calculate the following without actually multiplying the big numbers together:
(a) (23 · 99) mod 100
(b) (15 · 58) mod 60
(c) (54 · 59) mod 60
(d) (1287 · 1286) mod 1288
12. Determine whether each of the following numbers is divisible by 3: 56, 100, 143,
144, 89435640, 222222, 2222222
13. Determine whether each of the following numbers is divisible by 9: 2127, 2137,
2147, 2157, 2167, 2177, 2187, 2197
14. Find a number between 3500 and 3600 ending in 4 which is divisible by 9.
15. Determine whether each of the following numbers is divisible by 11: 6578, 4297,
1001, 10001, 100001, 1000001
16. Find a number between 2800 and 2900 ending in 8 that is divisible by 11.
17. Decide if each of the following numbers are prime or not: 111, 121, 131, 141
18. Factor 1800 into a product of primes.
19. Find the greatest common divisor of each of the following numbers using the
Euclidean Algorithm. Write the gcd as a linear combination of the two numbers.
Math 202 Homework Set A
2.2.2007
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(a) 1150 and 1845
(b) 38 and 16
(c) 84 and 72
(d) 4655 and 12075
(e) 1369 and 2597
20. Find multiplicative inverses or state that they do not exist:
(a) 5−1 mod 12
(b) 8−1 mod 18
(c) 9−1 mod 10
(d) 13−1 mod 60
(e) 28−1 mod 49
21. Solve the following congruences. Give all solutions; if there is no solution then
say so:
(a) 5x ≡ 3 mod 10
(b) 3x ≡ 7 mod 10
(c) 7x + 6 ≡ 4 mod 8
(d) 4x + 3 ≡ 10 mod 20
(e) 4x ≡ 6 mod 10
(f) 33x ≡ 22 mod 105
22. Cast out 9s in the following problems. Indicate any solutions which must be
wrong based on the method of casting out 9s.
(a) 5497 + 6949 = 12456
(b) 6237 + 9467 − 2684 = 13020
(c) 58647 · 68512 = 4081023264
(d) 265 · 95467 = 25288755
23. A number is called a perfect square if it is equal to an integer multiplied by
itself. Examples of perfect squares include 1, 4, 9, 16, 25, 36, 49, ... What are the
possible remainders when a perfect square is divided by 4?
24. Explain why no perfect square can end with the digits 79.
25. Explain in your own words what it means for two positive integers to be congruent mod 10.
Math 202 Homework Set A
2.2.2007
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26. Explain in your own words what it means for two positive integers to be congruent mod 100.
27. Explain in your own words what it means for two positive integers to be congruent mod 2.
28. Explain why, when reducing a number modulo 5, only the last digit matters.
29. Explain why, when reducing a number modulo 4, only the last digit matters.
30. If July 4, 2006 falls on a Tuesday, what day of the week does July 4, 2007 fall
on?
31. If January 1, 2010 falls on a Monday, what day of the week does January 1,
5010 fall on (assuming that there are 678 leap years between 2010 and 5010)?