MathJewels math guide · Explain factorization uniqueness up to units and associates
Units, Associates, and Factorization
Identify the integer units 1 and -1, recognize factors that differ by a unit, and understand the exact qualifiers in uniqueness of factorization.
The big idea
A unit is an integer that has an integer multiplicative inverse. The only integer units are \(1\) and \(-1\). Multiplying by 1 changes nothing, while multiplying by \(-1\) changes only a sign. Two integers are associates when one is a unit times the other. This is why a uniqueness statement about integer factorization must allow factors to be reordered and replaced by associates.
Worked example
Compare
\[ 42=2\cdot3\cdot7 \]
with
\[ 42=(-2)\cdot3\cdot(-7). \]
The lists are not literally equal after reordering, but \(-2=(-1)2\), \(3=1\cdot3\), and \(-7=(-1)7\). Corresponding factors are associates, and the two extra signs cancel because \((-1)(-1)=1\).
To find every unit, suppose \(uv=1\) for integers \(u,v\). Then \(|u||v|=1\), forcing \(|u|=|v|=1\). Hence \(u\) is 1 or \(-1\).
How children may show it
A learner may circle unit factors, pair signed factors with arrows labeled \(\times(-1)\), or place two factorizations in columns and match associates. The representation should distinguish a changed sign from a genuinely different irreducible factor.
Common mix-up
Learners sometimes call 1 prime or insert unlimited copies of 1 into a “prime factorization.” Ask, “Does this factor stop a nonunit factorization, or can it be removed without changing the number?” Units are handled separately precisely because they can be inserted or moved harmlessly.
Try it together
Write three irreducible factorizations of \(-60\) using different orders and sign placements. Match corresponding factors by associates, and verify that the total number of negative signs gives the correct sign of the product.