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Algebra

Abstract Algebra

Abstract Algebra Warmup

         

What real number does not have a multiplicative inverse?

Definition. A real number \(b\) is called the multiplicative inverse of a if

\[a \times b = b \times a = 1.\]

What number \(a\) satisfies \[3 + a \equiv 0 \text{ (mod 7)?}\]

Definition. Addition modulo 7 is defined by adding two numbers, dividing the sum by 7, and taking the remainder.

For example, \(3 + 6 \equiv 2\) (mod 7), since 3 + 6 = 9 and the remainder when 9 is divided by 7 is 2.

What number \(a\) satisfies \[3 \times a \equiv 1 \text{ (modulo 7)?}\]

Definition. Multiplication modulo 7 is defined by multiplying two numbers, dividing the product by 7, and taking the remainder.

For example, \(3 \times 6 \equiv 4\) (mod 7), since \(3 \times 6 = 18, \) and the remainder when 18 is divided by 7 is 4.

For which value of \(n\) does \[5 \times a \equiv 1 \text{ (mod n)}\]

have a solution?

Definition. Multiplication modulo n is defined by multiplying two numbers, dividing the product by n, and taking the remainder.

For example, take \(n = 6.\) We have \(5 \times 5 \equiv 1\) (mod 6), since \(5 \times 5 = 25, \) and the remainder when 25 is divided by 6 is 1.

Does every real number have an additive inverse?

Definition. A real number \(b\) is called the additive inverse of \(a\) if

\[a + b = b + a = 0.\]

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