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From pure math to modern cryptography to organic chemistry, abstract algebra underpins much of contemporary research. Learn the theory of groups, rings, and fields.

What real number does not have a multiplicative inverse?

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

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

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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.

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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.

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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.

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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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