Concept 1.1
The Fundamental Theorem of Arithmetic
Source: NCERT Mathematics, Class 108 minute lesson
Formal statement
Every composite number can be expressed (factorised) as a product of primes, and this factorisation is unique, apart from the order in which the prime factors occur.
The intuition
Primes are the atoms of multiplication. However you start breaking a number down, 360 = 4 × 90 or 360 = 10 × 36, you end with the same atoms: three 2s, two 3s and one 5.
That is the surprising part: the route doesn't matter, the destination does. So a factorisation is only finished when nothing in it can be split any further.
The attention
Look at the leaves of the factor tree, not the branches.
360 = 2 × 2 × 2 × 3 × 3 × 5 not 4 × 90
Look hereNot here
- Look here
- The prime factors, the leaves. Only the primes at the end are fixed. They are the number's fingerprint, the same whichever way you split.
- Not here
- A split into composite numbers. 4 × 90 is a true split, but it isn't finished. 4 and 90 can still be broken, so they tell you nothing unique about 360.
- What it teaches
- When a process can take many routes, look for what every route agrees on.