Properties of Congruence: Addition and Multiplication with Moduli

1) If a ≡ b (mod n), then which is always true?

A - b is divisible by a:

A + b is divisible by n:

A × b is divisible by n:

A - b is divisible by n:

The statement a ≡ b (mod n) means that the difference a - b is divisible by n. Therefore, option D is always true, as it directly follows from the definition of congruence modulo n.

Explanation

The statement a ≡ b (mod n) means that the difference a - b is divisible by n. Therefore, option D is always true, as it directly follows from the definition of congruence modulo n.

2) If 12 2 (mod 5), what is 2 d7 12 mod 5?

2

3

4

0

To solve for 2 × 12 mod 5, we first calculate 12 mod 5, which equals 2 (since 12 = 5 × 2 + 2). Now we multiply 2 by 2 to get 4. Finally, we find 4 mod 5, which is 4. Hence, the answer is 4.

Explanation

To solve for 2 × 12 mod 5, we first calculate 12 mod 5, which equals 2 (since 12 = 5 × 2 + 2). Now we multiply 2 by 2 to get 4. Finally, we find 4 mod 5, which is 4. Hence, the answer is 4.

3) If x 4 (mod 6), what is 3x mod 6?

1

0

2

4

To solve for 3x mod 6, we first note that x is congruent to 4 modulo 6. Thus, we can substitute x with 4: 3(4) mod 6 = 12 mod 6 = 0. Therefore, the correct answer is 0, which corresponds to option B.

Explanation

To solve for 3x mod 6, we first note that x is congruent to 4 modulo 6. Thus, we can substitute x with 4: 3(4) mod 6 = 12 mod 6 = 0. Therefore, the correct answer is 0, which corresponds to option B.

4) Which is equivalent: 45 3 ? (mod 9)

1

3

0

6

To find 45 mod 9, we divide 45 by 9, which gives us a quotient of 5 and a remainder of 0. Therefore, 45 is equivalent to 0 modulo 9.

Explanation

To find 45 mod 9, we divide 45 by 9, which gives us a quotient of 5 and a remainder of 0. Therefore, 45 is equivalent to 0 modulo 9.

5) If 8 ≡ 2 (mod 3), then 4 × 8 ≡ ? (mod 3)

0

1

2

3

To solve for 4 × 8 (mod 3), we first utilize the given information that 8 is congruent to 2 modulo 3. Therefore, we can substitute 8 with 2: 4 × 8 (mod 3) becomes 4 × 2 (mod 3). Calculating this yields 8, and when we take 8 modulo 3, we find that 8 is congruent to 2 (since 8 divided by 3 leaves a remainder of 2). Thus, 4 × 8 ≡ 2 (mod 3).

Explanation

To solve for 4 × 8 (mod 3), we first utilize the given information that 8 is congruent to 2 modulo 3. Therefore, we can substitute 8 with 2: 4 × 8 (mod 3) becomes 4 × 2 (mod 3). Calculating this yields 8, and when we take 8 modulo 3, we find that 8 is congruent to 2 (since 8 divided by 3 leaves a remainder of 2). Thus, 4 × 8 ≡ 2 (mod 3).

6) If 100 ≡ 4 (mod 12), what is 2 × 100 mod 12?

2

4

8

10

To solve for 2 × 100 mod 12, we first recognize that 100 is congruent to 4 modulo 12. This means we can replace 100 with 4 in our calculation. Therefore, 2 × 100 mod 12 is equivalent to 2 × 4 mod 12, which equals 8 mod 12. Since 8 is less than 12, the result is simply 8. Hence, the correct answer is C (8).

Explanation

To solve for 2 × 100 mod 12, we first recognize that 100 is congruent to 4 modulo 12. This means we can replace 100 with 4 in our calculation. Therefore, 2 × 100 mod 12 is equivalent to 2 × 4 mod 12, which equals 8 mod 12. Since 8 is less than 12, the result is simply 8. Hence, the correct answer is C (8).

7) If 39 ≡ 3 (mod 6), which is also true?

46 ≡ 3 (mod 6)

47 ≡ 3 (mod 6)

45 ≡ 3 (mod 6)

49 ≡ 3 (mod 6)

To determine which statement is true, we can evaluate each option based on modular arithmetic. Since 39 gives a remainder of 3 when divided by 6, we can calculate the remainders of the other numbers when divided by 6: 46 gives a remainder of 4, 47 gives a remainder of 5, and 49 gives a remainder of 1. However, 45 gives a remainder of 3, making option C the correct choice.

Explanation

To determine which statement is true, we can evaluate each option based on modular arithmetic. Since 39 gives a remainder of 3 when divided by 6, we can calculate the remainders of the other numbers when divided by 6: 46 gives a remainder of 4, 47 gives a remainder of 5, and 49 gives a remainder of 1. However, 45 gives a remainder of 3, making option C the correct choice.

8) Which is a valid congruence?

19 ≡ 2 (mod 7)

19 ≡ 03 (mod 7)

19 ≡ 05 (mod 7)

19 ≡ 06 (mod 7)

To determine the validity of the congruences, we can calculate 19 modulo 7. The result of 19 divided by 7 gives a remainder of 5. Therefore, the congruence 19 ≡ 5 (mod 7) is valid, making option C the correct answer. The other options do not hold true since they do not match the remainder of 5.

Explanation

To determine the validity of the congruences, we can calculate 19 modulo 7. The result of 19 divided by 7 gives a remainder of 5. Therefore, the congruence 19 ≡ 5 (mod 7) is valid, making option C the correct answer. The other options do not hold true since they do not match the remainder of 5.

9) If a ≡ b (mod n), then for any integer k:

Ka ≡ kb (mod k)

Ka + n ≡ kb (mod n)

Ka ≡ kb (mod n)

Ka - kb ≡ 1 (mod n)

The expression a ≡ b (mod n) means that the difference a - b is divisible by n. When you multiply both sides of the congruence by an integer k, it preserves the congruence relation, leading to ka ≡ kb (mod n). This holds true because if a - b is divisible by n, then k(a - b) is also divisible by n.

Explanation

The expression a ≡ b (mod n) means that the difference a - b is divisible by n. When you multiply both sides of the congruence by an integer k, it preserves the congruence relation, leading to ka ≡ kb (mod n). This holds true because if a - b is divisible by n, then k(a - b) is also divisible by n.

10) Which property does modular arithmetic share with normal integers?

Closure under addition

Closure under multiplication

Both A & B

Neither

Modular arithmetic shares the properties of closure under both addition and multiplication with normal integers. This means that the sum or product of two integers, when taken modulo a certain number, will also result in an integer within the same range. Hence, both operations maintain the result within the modular system.

Explanation

Modular arithmetic shares the properties of closure under both addition and multiplication with normal integers. This means that the sum or product of two integers, when taken modulo a certain number, will also result in an integer within the same range. Hence, both operations maintain the result within the modular system.