$76 \equiv -152 \pmod{3}$
Answer (2)
The congruence 76 ≡ − 152 ( mod 3 ) is true because both 76 and − 152 yield a remainder of 1 when divided by 3 . Therefore, they are equivalent in modulo 3 .
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Reduce 76 modulo 3 , which gives 76 ≡ 1 ( mod 3 ) .
Reduce − 152 modulo 3 , which gives − 152 ≡ 1 ( mod 3 ) .
Compare the two results.
Since both are congruent to 1 ( mod 3 ) , the congruence is True .
Explanation
Understanding the Congruence We are asked to verify the congruence 76 ≡ − 152 ( mod 3 ) . This means we need to check if 76 and − 152 have the same remainder when divided by 3 .
Reducing 76 modulo 3 First, let's find the remainder of 76 when divided by 3 . We can perform the division 76 ÷ 3 . The result is 25 with a remainder of 1 . Therefore, 76 ≡ 1 ( mod 3 ) .
Reducing -152 modulo 3 Next, let's find the remainder of − 152 when divided by 3 . We can perform the division − 152 ÷ 3 . The result is − 51 with a remainder of 1 . Therefore, − 152 ≡ 1 ( mod 3 ) .
Comparing the Results Since 76 ≡ 1 ( mod 3 ) and − 152 ≡ 1 ( mod 3 ) , we can conclude that 76 ≡ − 152 ( mod 3 ) is true.
Final Answer Therefore, the congruence 76 ≡ − 152 ( mod 3 ) is true. True
Examples
Modular arithmetic is used in cryptography to ensure secure communication. For example, the RSA algorithm relies on the properties of modular arithmetic to encrypt and decrypt messages. In daily life, it can be used to determine the day of the week for a future date. If today is Wednesday, which is day 3, then in 30 days it will be day 3 + 30 ≡ 33 ≡ 5 ( mod 7 ) , which is a Friday.
