Every Number Has Exactly One Blueprint Real Numbers

Sonia drives one round of a circular sports field track in 18 minutes; Ravi drives one round of the same track in 12 minutes. Both start at the same point at the same time, going the same direction. After how many minutes will they next both be back at the starting point together? Sonia returns to the start at 18, 36, 54, 72... minutes; Ravi returns at 12, 24, 36, 48... minutes. The first number appearing in both lists is 36. This chapter builds the tool that finds that number directly, without listing multiples by hand: the Lowest Common Multiple, and its close relative the Highest Common Factor -- both built from a single underlying fact about how numbers are put together.

Take any composite number and break it down into a factor tree, splitting it again and again until only prime numbers remain: 32760 breaks down to 2 x 2 x 2 x 3 x 3 x 5 x 7 x 13, or written more compactly using powers, 2 cubed x 3 squared x 5 x 7 x 13. Try building that same factor tree a different way, splitting the number differently at each step, and the exact same collection of prime factors appears every time, just possibly in a different order of discovery. This is the Fundamental Theorem of Arithmetic: every composite number can be factorised into primes, and that factorisation is unique, except for the order the primes are written in. An equivalent result appears as far back as Book IX, Proposition 14 of Euclid's Elements, though the first fully rigorous proof of it is credited to Carl Friedrich Gauss, sometimes called the 'Prince of Mathematicians', in his 1801 book Disquisitiones Arithmeticae. This uniqueness is not just a curiosity -- it directly proves useful facts. Since 4 to the power n always factorises as 2 to the power (2n), and nothing else, the digit 5 can never appear anywhere in its prime factorisation, however large n gets; since a number can only end in the digit 0 if its factorisation includes both a 2 and a 5, the Fundamental Theorem of Arithmetic guarantees, with total certainty, that 4 to the power n can never end in the digit 0, for any natural number n at all.

With every number's blueprint of prime powers laid out, HCF and LCM stop needing lists of multiples entirely. HCF is the product of the smallest power of each prime that is common to every number involved; LCM is the product of the greatest power of each prime that appears in any of them. For 6 = 2 x 3 and 20 = 2 squared x 5: the only shared prime is 2, at its smallest shared power (2 to the power 1), giving HCF = 2; the LCM takes the greatest power of every prime involved, 2 squared x 3 x 5 = 60. For 96 = 2 to the power 5 x 3 and 404 = 2 squared x 101: HCF = 2 squared = 4, and since LCM x HCF must equal the product of the two numbers, LCM = (96 x 404)/4 = 9696. A clean identity falls out of this every time two numbers are involved: HCF(a,b) x LCM(a,b) = a x b, useful precisely because it lets one of HCF or LCM be found directly from the other without repeating the factorisation work. This identity has a real limit worth remembering: for 6, 72, and 120 together, HCF = 6 and LCM = 360, but 6 x 72 x 120 does not equal 6 x 360 -- the neat two-number identity simply does not extend to three or more numbers. A more complicated, but still exact, replacement exists for three numbers: LCM(p,q,r) = (pqr x HCF(p,q,r)) / (HCF(p,q) x HCF(q,r) x HCF(p,r)), with the matching formula for HCF(p,q,r) built the same way from pairwise LCMs instead.

The Fundamental Theorem of Arithmetic has one more use in this chapter, and it is a genuinely different kind of argument: proving that certain numbers can never be written as a fraction at all. First, a lemma: if a prime p divides a perfect square a squared, then p must also divide a itself. This follows directly from uniqueness -- factorise a into primes, square it, and the only primes that can possibly divide that square are the same primes that were already in a to begin with; there is no way for a brand new prime to sneak in. Now assume, for the sake of argument, that the square root of 2 COULD be written as a fraction r/s in lowest terms (r and s sharing no common factor). Squaring both sides gives 2 = r squared / s squared, so r squared = 2 s squared, meaning 2 divides r squared, and by the lemma, 2 must divide r itself. Write r = 2c. Substituting back: 4c squared = 2 s squared, so s squared = 2c squared, meaning 2 divides s squared too, and so 2 divides s as well. But then both r and s share a factor of 2, directly contradicting the assumption that they had no common factor at all. The only way out of this contradiction is that the original assumption was false: the square root of 2 cannot be written as a fraction. It is irrational. The exact same argument, with 3 in place of 2 throughout, proves the square root of 3 is irrational too, and the same technique generalises to the square root of any prime number.

A short list of closure facts, each provable the same way (assume the opposite, reach a contradiction), turns single irrational proofs into proofs about whole families of related numbers: the sum or difference of a rational number and an irrational number is always irrational, and the product or quotient of a nonzero rational number with an irrational number is always irrational. To show 5 minus the square root of 3 is irrational: assume it equals some fraction a/b. Rearranging gives the square root of 3 equals 5 minus a/b, a combination of two rational numbers, which would itself have to be rational -- directly contradicting the already-proven fact that the square root of 3 is irrational. The same contradiction-via-rearranging trick shows 3 times the square root of 2 is irrational: assume it equals a/b, rearrange to get the square root of 2 equals a/(3b), again forcing an irrational number to equal a ratio of integers. Every combination of a rational with a proven irrational, built this way, inherits the same contradiction.

Sonia and Ravi's meeting time, 36 minutes, comes from exactly the same tool -- prime factorisation -- that also proved the square root of 2 can never be written as a fraction: 18 = 2 x 3 squared and 12 = 2 squared x 3, so LCM = 2 squared x 3 squared = 36. Two halves of one chapter, HCF/LCM on one side and irrationality proofs on the other, look like unrelated topics stitched together, until it's clear both genuinely run on the same single fact: every number's prime factorisation is unique. One direction of that uniqueness builds formulas for HCF and LCM directly from the blueprint; the other direction is exactly what makes the contradiction bite in every irrationality proof in this chapter. Worth flagging honestly: this book's own opening paragraph mentions Euclid's division algorithm as something 'we begin with... in Section 1.2' -- but the section that actually follows covers only the Fundamental Theorem of Arithmetic, with no division-algorithm content at all. That is not a misprint to chase down; the current syllabus no longer includes Euclid's Division Lemma or decimal-expansion classification in this chapter (both were part of older editions), and the intro sentence is simply a leftover reference from before that trim.