False statementConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passverified 2026-08-02 (claude-opus-5)

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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FALSE: For all integers $a$ and $b$ , $\gcd(a,b) \cdot \operatorname{lcm}(a,b) = ab$

Statement

False claim: for all integers $a$ and $b$ ,

$\gcd(a,b) \cdot \operatorname{lcm}(a,b) \;=\; ab$

(Common divisor, and the greatest common divisor $\gcd(a,b)$ , with the convention $\gcd(0,0) := 0$ , Common multiple, and the least common multiple $\operatorname{lcm}(a,b)$ , taken to be $0$ when $a = 0$ or $b = 0$ ).

The true statement is Every common multiple of $a$ and $b$ is a multiple of $\operatorname{lcm}(a,b)$ , and $\gcd(a,b) \cdot \operatorname{lcm}(a,b) = |ab|$ , with an absolute value on the right: $\gcd(a,b)\operatorname{lcm}(a,b) = |ab|$ . The two differ as soon as $ab$ is negative, and $(a,b) = (-2,3)$ is a witness: there $\gcd(-2,3) = 1$ and $\operatorname{lcm}(-2,3) = 6$ , so the left side is $6$ , while $ab = -6$ .

Numerals. For $k \in \mathbb{N}$ the symbol $k$ inside $\mathbb{Z}$ means $\iota(k)$ , the embedding of The naturals embed in the integers.

Facts & Assumptions

Given: The integers $-2$ , $3$ and $6$ .

[L1]

$\mathbb{Z}$ is a commutative ring: multiplication is associative and commutative, $x \cdot 1 = x$ , $x \cdot (-1) = -x$ , multiplication distributes over addition, and every $x$ has an additive inverse (The integers form a commutative ring, Arithmetic on the integers, The integers as equivalence classes of pairs of naturals).

[L2]

The order on $\mathbb{Z}$ is total, antisymmetric and transitive and is compatible with addition (The integers form a totally ordered ring, Order on the integers).

[L3]

$\gcd(a,b)\operatorname{lcm}(a,b) = |ab|$ for all $a, b$ (Every common multiple of $a$ and $b$ is a multiple of $\operatorname{lcm}(a,b)$ , and $\gcd(a,b) \cdot \operatorname{lcm}(a,b) = |ab|$ , Common multiple, and the least common multiple $\operatorname{lcm}(a,b)$ , taken to be $0$ when $a = 0$ or $b = 0$ ).

[L4]

$d = \gcd(u,v)$ exactly when $d \ge 0$ , $d \mid u$ , $d \mid v$ , and every common divisor of $u$ and $v$ divides $d$ (Every common divisor of $a$ and $b$ divides $\gcd(a,b)$ ; consequently $d = \gcd(a,b)$ exactly when $d \ge 0$ , $d \mid a$ , $d \mid b$ , and every common divisor of $a$ and $b$ divides $d$ — a characterisation that holds at $(a,b) = (0,0)$ as well).

[L5]

If $d \mid u$ and $d \mid v$ then $d \mid ux + vy$ for all $x, y$ ; and $1 \mid u$ for every $u$ (Divisibility is reflexive and transitive on $\mathbb{Z}$ , and is linear: if $d \mid a$ and $d \mid b$ then $d \mid ax + by$ for all integers $x, y$ ; also $d \mid a$ implies $d \mid ac$ , $-d \mid a$ and $d \mid -a$ , Divisibility in $\mathbb{Z}$ : $d \mid a$ when $a = dq$ for some integer $q$ ).

[L6]

$|x| = x$ for $x \ge 0$ and $|x| = -x$ for $x \le 0$ ; $|x| \ge 0$ (The absolute value $|a|$ of an integer, Absolute value in $\mathbb{Z}$ : $|a| \ge 0$ ; $|a| = 0$ exactly when $a = 0$ ; $|-a| = |a|$ ; $|ab| = |a|\,|b|$ ; $-|a| \le a \le |a|$ ; and $|a| \le c$ exactly when $-c \le a \le c$ ).

[L7]

If $xz = yz$ and $z \ne 0$ then $x = y$ (The integers have no zero divisors; multiplicative cancellation).

[L8]

$\iota$ is injective with image the nonnegative integers, and preserves addition and multiplication (The naturals embed in the integers).

[L9]

The refuted claim: $\gcd(a,b)\operatorname{lcm}(a,b) = ab$ for all integers $a, b$ .

Refutation

technique · direct

1.1

$\gcd(-2,3) = 1$ . Indeed $1 \ge 0$ , and $1 \mid -2$ and $1 \mid 3$ by [L5]; and $(-2) \cdot 1 + 3 \cdot 1 = 1$ , so every common divisor of $-2$ and $3$ divides $1$ by [L5]. By [L4] this characterises $\gcd(-2,3)$ .

L1L4L5L8algebra

1.2

$(-2) \cdot 3 = -6$ , and $|-6| = 6$ since $-6 \le 0$ and $-(-6) = 6$ .

L1L2L6L8algebra

1.3

$6 \ne -6$ : otherwise $6 + 6 = 0$ , that is $\iota(12) = 0 = \iota(0)$ , whence $12 = 0$ in $\mathbb{N}$ , contradicting injectivity of $\iota$ .

L1L8algebra

2.1

By [L3], $\gcd(-2,3)\operatorname{lcm}(-2,3) = |(-2) \cdot 3| = 6$ ; with step 1.1 this reads $1 \cdot \operatorname{lcm}(-2,3) = 6$ , so $\operatorname{lcm}(-2,3) = 6$ by cancellation of the nonzero factor $1$ .

step 1.1step 1.2L1L3L7

3.1

Therefore $\gcd(-2,3)\operatorname{lcm}(-2,3) = 1 \cdot 6 = 6$ , while $(-2) \cdot 3 = -6$ , and these differ: [L9] is false at $(a,b) = (-2,3)$ .

step 1.1step 1.2step 2.1step 1.3L1L9∎

Remarks

Every pair with $ab < 0$ refutes it, not just this one: $\gcd$ and $\operatorname{lcm}$ are both nonnegative by construction (Common divisor, and the greatest common divisor $\gcd(a,b)$ , with the convention $\gcd(0,0) := 0$ , Common multiple, and the least common multiple $\operatorname{lcm}(a,b)$ , taken to be $0$ when $a = 0$ or $b = 0$ ), so their product is nonnegative, while $ab$ is negative. The witness above is simply a small instance.
The claim is true when $a$ and $b$ are both nonnegative, which is why it is a natural slip: in that case $|ab| = ab$ and the two statements coincide. Every common multiple of $a$ and $b$ is a multiple of $\operatorname{lcm}(a,b)$ , and $\gcd(a,b) \cdot \operatorname{lcm}(a,b) = |ab|$ is the version that holds for all integers.
The absolute value is not the only convention doing work. At $a = 0$ the true identity reads $\gcd(0,b) \cdot 0 = 0 = |0 \cdot b|$ , which holds only because $\operatorname{lcm}(0,b) = 0$ was fixed in Common multiple, and the least common multiple $\operatorname{lcm}(a,b)$ , taken to be $0$ when $a = 0$ or $b = 0$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 63 results over 23 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

Least common multiple (Wikipedia) (standard reference, not scraped)
Greatest common divisor (Wikipedia) (standard reference, not scraped)

FALSE: For all integers aaa and bbb, gcd⁡(a,b)⋅lcm⁡(a,b)=ab\gcd(a,b) \cdot \operatorname{lcm}(a,b) = abgcd(a,b)⋅lcm(a,b)=ab