Linear Congruence Quiz: Solve Modular Equations

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Quizzes Created: 7682 | Total Attempts: 9,547,133
| Questions: 20 | Updated: Dec 16, 2025
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1) What is the last digit of 7^23?

Explanation

Powers of 7 mod 10 repeat as 7, 9, 3, 1 every 4. 23 mod 4 = 3 → 3rd term → 3.

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About This Quiz
Linear Congruence Quiz: Solve Modular Equations - Quiz

Linear congruences may look basic, but they lead to some interesting number patterns. This quiz shows how these equations work when you solve them using modulo and how the answers fit into neat, repeating cycles. As you move through the questions, you’ll see why this idea is so useful in... see moremath and how quickly the patterns start to make sense. Give it a try and enjoy solving these modular puzzles!
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2) 2^50 mod 5 equals

Explanation

Cycle: 2, 4, 3, 1 (length 4). 50 mod 4 = 2 → 2nd term → 4.

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3) 3^40 mod 7 equals

Explanation

Cycle repeats every 6. 40 mod 6 = 4. 3^4 = 81 ≡ 81 − 77 = 4 (mod 7).

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4) For any positive integer n, 9^n ends in 9 when n is odd and ends in 1 when n is even.

Explanation

Mod 10 cycle for 9 is 9, 1, 9, 1 → alternates with n.

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5) 4^15 mod 3 = ___

Explanation

4 ≡ 1 (mod 3). So 4^15 ≡ 1^15 = 1.

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6) Select all numbers congruent to 2^13 mod 7.

Explanation

2^3 = 8 ≡ 1 (mod 7). Then 2^13 ≡ 2 (mod 7). Numbers ≡ 2 mod 7 are 9, 16, 23, 30.

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7) 5^123 mod 10 equals

Explanation

Any power of 5 ends in 5 → mod 10 = 5.

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8) 6^98 mod 4 equals

Explanation

6 ≡ 2 (mod 4). For n ≥ 2, 2^n is multiple of 4 → 0 mod 4.

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9) 11^20 mod 11 = ___

Explanation

11 ≡ 0 (mod 11). So 0^20 = 0.

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10) 7^0 mod 9 =

Explanation

Any nonzero number to the 0th power is 1. 1 mod 9 = 1.

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11) 3^100 ≡ 1 (mod 3).

Explanation

3 ≡ 0 (mod 3). So 3^100 ≡ 0 mod 3.

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12) 8^25 mod 5 equals

Explanation

8 ≡ 3 (mod 5). Cycle: 3, 4, 2, 1. 25 mod 4 = 1 → 1st term → 3.

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13) 10^n mod 6 for any positive integer n equals

Explanation

10 ≡ 4 (mod 6). 4^1 ≡ 4, 4^2 ≡ 4 → always 4.

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14) 12^37 mod 9 = ___

Explanation

12 ≡ 3 (mod 9). 3^2 ≡ 0 (mod 9) → for any exponent ≥ 2, result 0.

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15) Select all true statements.

Explanation

2 ≡ −1 (mod 3). (−1)^20=1; (−1)^15=−1≡2; (−1)^6=1.

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16) 13^45 mod 10 equals

Explanation

13 ends in 3. Cycle: 3, 9, 7, 1. 45 mod 4 = 1 → 1st term → 3.

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17) 2^100 mod 8 = ___

Explanation

2^3 = 8 ≡ 0 (mod 8). Any higher power stays 0 mod 8.

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18) For any integers n ≥ 1 and m ≥ 2, 1^n ≡ 1 (mod m).

Explanation

1^n = 1 for all n → remainder 1.

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19) 7^14 mod 5 equals

Explanation

7 ≡ 2 (mod 5). Cycle: 2, 4, 3, 1. 14 mod 4 = 2 → 2nd term → 4.

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20) 9^23 mod 7 equals

Explanation

9 ≡ 2 (mod 7). Powers of 2 mod 7: 2, 4, 1. 23 mod 3 = 2 → 2nd term → 4.

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What is the last digit of 7^23?
2^50 mod 5 equals
3^40 mod 7 equals
For any positive integer n, 9^n ends in 9 when n is odd and ends in 1...
4^15 mod 3 = ___
Select all numbers congruent to 2^13 mod 7.
5^123 mod 10 equals
6^98 mod 4 equals
11^20 mod 11 = ___
7^0 mod 9 =
3^100 ≡ 1 (mod 3).
8^25 mod 5 equals
10^n mod 6 for any positive integer n equals
12^37 mod 9 = ___
Select all true statements.
13^45 mod 10 equals
2^100 mod 8 = ___
For any integers n ≥ 1 and m ≥ 2, 1^n ≡ 1 (mod m).
7^14 mod 5 equals
9^23 mod 7 equals
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