12 ≡ 15 mod 4&nbsp;8 ≡ 21 mod 3&nbsp;25 ≡ 60 mod 7&nbsp;62 ≡ 30 mod 5&nbsp;35 ≡ 95 mod 12&nbsp;

To solve the given congruences, we'll determine whether the left-hand side (LHS) is congruent to the right-hand side (RHS) modulo the given modulus.(a) 12≡15(mod4)To check if 12 is congruent to 15 modulo 4, we need to verify if their difference is divisible by 4. Let's calculate:15−12=3Since 3 is not divisible by 4, we conclude that 12 is not congruent to 15 modulo 4.(b) 8≡21(mod3)Again, we'll check if the difference between 8 and 21 is divisible by 3:21−8=13As 13 is not divisible by 3, we can determine that 8 is not congruent to 21 modulo 3.(c) 25≡60(mod7)Similar to the previous steps, we'll find the difference between 25 and 60:60−25=35Since 35 is not divisible by 7, we conclude that 25 is not congruent to 60 modulo 7.(d) 62≡30(mod5)Calculating the difference between 62 and 30:62−30=32As 32 is divisible by 5, we can determine that 62 is congruent to 30 modulo 5.(e) 35≡95(mod12)The difference between 35 and 95 is:95−35=60Since 60 is divisible by 12, we conclude that 35 is congruent to 95 modulo 12.To summarize the results:(a) 12≢15(mod4)(b) 8≢21(mod3)(c) 25≢60(mod7)(d) 62≡30(mod5)(e) 35≡95(mod12)

"12 ≡ 15 mod 48 ≡ 21 mod..." solved and explained

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Answered question

2022-05-15

12 ≡ 15 mod 4

8 ≡ 21 mod 3

25 ≡ 60 mod 7

62 ≡ 30 mod 5

35 ≡ 95 mod 12

Answer & Explanation

Mr Solver

Skilled2023-05-14Added 147 answers

To solve the given congruences, we'll determine whether the left-hand side (LHS) is congruent to the right-hand side (RHS) modulo the given modulus.
(a)

12 \equiv 15 (\mod 4)

To check if 12 is congruent to 15 modulo 4, we need to verify if their difference is divisible by 4. Let's calculate:

15 - 12 = 3

Since 3 is not divisible by 4, we conclude that 12 is not congruent to 15 modulo 4.
(b)

8 \equiv 21 (\mod 3)

Again, we'll check if the difference between 8 and 21 is divisible by 3:

21 - 8 = 13

As 13 is not divisible by 3, we can determine that 8 is not congruent to 21 modulo 3.
(c)

25 \equiv 60 (\mod 7)

Similar to the previous steps, we'll find the difference between 25 and 60:

60 - 25 = 35

Since 35 is not divisible by 7, we conclude that 25 is not congruent to 60 modulo 7.
(d)

62 \equiv 30 (\mod 5)

Calculating the difference between 62 and 30:

62 - 30 = 32

As 32 is divisible by 5, we can determine that 62 is congruent to 30 modulo 5.
(e)

35 \equiv 95 (\mod 12)

The difference between 35 and 95 is:

95 - 35 = 60

Since 60 is divisible by 12, we conclude that 35 is congruent to 95 modulo 12.
To summarize the results:
(a)

12 ≢ 15 (\mod 4)

(b)

8 ≢ 21 (\mod 3)

(c)

25 ≢ 60 (\mod 7)

(d)

62 \equiv 30 (\mod 5)

(e)

35 \equiv 95 (\mod 12)

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