Discrete Math Test Questions with Correct
Answers
Arrange the following steps into a proof that if a | b and a | c, then a | (b + c).
1. We will use a direct proof
2. Suppose that a | b and a | c
3. Then it follows that there must exist integers s and t with b=a-s and c=a*t
4. Hence, b+c=as+at=a(s+t)
5. Therefore, a | b+c
Match the pair of integers a and d > 0 with the unique integers q and r with 0
≤ r < d such that a = d⋅q + r.
1.a = 80, d = 9
2. a = –40, d = 7
3. a = 99, d = 11
4. a = 0, d = 13
5. a = 51, d = 6
1. q=8, r=8
2. q=-6 , r-2
3. q=9 , r=0
4. q=0 , r=0
5. q=8 , r=3
Put these statements in order to produce a proof that the integers a and b are congruent
modulo m if and only if there is an integer k such that a = b + km.
,1. We Perform a direct proof in both directions, first proving sufficiency and then necessity
2. if a= b (mod m) then m | (a-b)
3. Then there is an integer k such that km=a-b, so that a=b+km
4. Now, we perform a direct proof of the other direction, beginning with the assumption that
there is an integer k such that a=b+km
5. If there is an integer k such that a=b+km, then km=a-b
6. Hence, m divides a-b, so that a=b(mod m).
Which of these are true, where m is a positive integer and a, b, c, and d are integers?
1. (a + b) mod m = ((a mod m) + (b mod m)) mod m
2. If a ≡ b (mod m) and c ≡ d (mod m), then ac ≡ bd (mod m).
3. If a ≡ b (mod m) and c ≡ d (mod m), then a + c ≡ b + d (mod m).
Match each integer with one that divides it.
Instructions
18 -----------------------------------9
-14-----------------------------------7
11------------------------------------11
65----------------------------------13
Match each expression involving modular arithmetic on the left with its value on the
right.
(16^2 mod 20) mod 7
2
(16^2 mod 20)^2 mod 7
,4
(5^2 mod 7)^3 mod 9
1
(5 mod 7)^2 mod 11
3
True or false: If a | b + 1 and a | c + 1, then a | b + c + 1 for all integers a, b, and c,
where a ≠0.
False
Which of these are true? (Select all that are correct.)
0 mod d = 0 for all d
0 div d = 0 for all d ≠ 0
a = d(a div d) + a mod d
d | a if and only if a mod d = 0
1. d | a if and only if a mod d = 0
2. a = d(a div d) + a mod d
3. 0 div d = 0 for all d ≠ 0
4. 0 mod d = 0 for all d
0 mod d = 0 for all d
0 div d = 0 for all d ≠ 0
a = d(a div d) + a mod d
, d | a if and only if a mod d = 0
Which of these is true where m is a positive integer and a and b are integers?
If a mod m = b mod m, then a ≡ b (mod m).
If a ≡ b (mod m), then a mod m = b mod m.
Given that 8 ≡ 3 (mod 5) and 9 ≡ 4 (mod 5), which of the following are true?
8 + 9 ≡ 3 + 4 (mod 5)
8 ⋅ 9 ≡ 3 ⋅ 4 (mod 5)
Match the property name on the left with the definition on the right.
Closure
If a and b belong to Zm, then a+mb and a ⋅mb belong to Zm.
If a and b belong to Zm, then a +m b and a ⋅mbbelong to Zm.
Associativity
If a, b and c belong to Zm, then (a+mb) +mc=a+m(b+mc) and (a⋅mb) ⋅mc=a⋅m(b⋅mc).
If a, b and c belong to Zm, then (a +m b) +m c = a +m (b +m c) and (a ⋅m b)
⋅m c = a ⋅m (b ⋅m c).
Commutativity
If a and b belong to Zm, then a+mb=b+maand a ⋅mb=b⋅ma.
If a and b belong to Zm, then a +m b = b +m a and a ⋅mb = b ⋅m a.
Identity elements
If a belongs to Zm, then a+m0=0 +ma=a and a⋅m1=1 ⋅ma=a.
Answers
Arrange the following steps into a proof that if a | b and a | c, then a | (b + c).
1. We will use a direct proof
2. Suppose that a | b and a | c
3. Then it follows that there must exist integers s and t with b=a-s and c=a*t
4. Hence, b+c=as+at=a(s+t)
5. Therefore, a | b+c
Match the pair of integers a and d > 0 with the unique integers q and r with 0
≤ r < d such that a = d⋅q + r.
1.a = 80, d = 9
2. a = –40, d = 7
3. a = 99, d = 11
4. a = 0, d = 13
5. a = 51, d = 6
1. q=8, r=8
2. q=-6 , r-2
3. q=9 , r=0
4. q=0 , r=0
5. q=8 , r=3
Put these statements in order to produce a proof that the integers a and b are congruent
modulo m if and only if there is an integer k such that a = b + km.
,1. We Perform a direct proof in both directions, first proving sufficiency and then necessity
2. if a= b (mod m) then m | (a-b)
3. Then there is an integer k such that km=a-b, so that a=b+km
4. Now, we perform a direct proof of the other direction, beginning with the assumption that
there is an integer k such that a=b+km
5. If there is an integer k such that a=b+km, then km=a-b
6. Hence, m divides a-b, so that a=b(mod m).
Which of these are true, where m is a positive integer and a, b, c, and d are integers?
1. (a + b) mod m = ((a mod m) + (b mod m)) mod m
2. If a ≡ b (mod m) and c ≡ d (mod m), then ac ≡ bd (mod m).
3. If a ≡ b (mod m) and c ≡ d (mod m), then a + c ≡ b + d (mod m).
Match each integer with one that divides it.
Instructions
18 -----------------------------------9
-14-----------------------------------7
11------------------------------------11
65----------------------------------13
Match each expression involving modular arithmetic on the left with its value on the
right.
(16^2 mod 20) mod 7
2
(16^2 mod 20)^2 mod 7
,4
(5^2 mod 7)^3 mod 9
1
(5 mod 7)^2 mod 11
3
True or false: If a | b + 1 and a | c + 1, then a | b + c + 1 for all integers a, b, and c,
where a ≠0.
False
Which of these are true? (Select all that are correct.)
0 mod d = 0 for all d
0 div d = 0 for all d ≠ 0
a = d(a div d) + a mod d
d | a if and only if a mod d = 0
1. d | a if and only if a mod d = 0
2. a = d(a div d) + a mod d
3. 0 div d = 0 for all d ≠ 0
4. 0 mod d = 0 for all d
0 mod d = 0 for all d
0 div d = 0 for all d ≠ 0
a = d(a div d) + a mod d
, d | a if and only if a mod d = 0
Which of these is true where m is a positive integer and a and b are integers?
If a mod m = b mod m, then a ≡ b (mod m).
If a ≡ b (mod m), then a mod m = b mod m.
Given that 8 ≡ 3 (mod 5) and 9 ≡ 4 (mod 5), which of the following are true?
8 + 9 ≡ 3 + 4 (mod 5)
8 ⋅ 9 ≡ 3 ⋅ 4 (mod 5)
Match the property name on the left with the definition on the right.
Closure
If a and b belong to Zm, then a+mb and a ⋅mb belong to Zm.
If a and b belong to Zm, then a +m b and a ⋅mbbelong to Zm.
Associativity
If a, b and c belong to Zm, then (a+mb) +mc=a+m(b+mc) and (a⋅mb) ⋅mc=a⋅m(b⋅mc).
If a, b and c belong to Zm, then (a +m b) +m c = a +m (b +m c) and (a ⋅m b)
⋅m c = a ⋅m (b ⋅m c).
Commutativity
If a and b belong to Zm, then a+mb=b+maand a ⋅mb=b⋅ma.
If a and b belong to Zm, then a +m b = b +m a and a ⋅mb = b ⋅m a.
Identity elements
If a belongs to Zm, then a+m0=0 +ma=a and a⋅m1=1 ⋅ma=a.
