Chapter 1.3 — Exercise 1.2 — Fundamental Theorem

The Fundamental Theorem of Arithmetic

Every composite number can be expressed as a product of primes, and that factorisation is unique — the order the factors are written in may differ, but the actual set of primes never does. Three people factorising 180 by starting from different primes all arrive at the same result:

This uniqueness is what turns prime factorisation into a genuine fingerprint for a number — one that you can use to compare numbers, spot common factors, and compute LCM with total precision, rather than by trial and error.

Five Numbers, Factorised

Question 1 asks for the prime factorisation of five numbers, using a factor tree or repeated division by primes:

The last two rows are worth a second look — 5005 and 7429 have no repeated prime factor at all, and neither is even or divisible by 3. Trying 2 or 3 first on numbers like these wastes time; if a number's digit sum isn't a multiple of 3 and it doesn't end in an even digit, jump straight to testing 5, 7, 11, and upward instead.

Building one of these by hand shows the process clearly. Starting from 140: it's even, so divide by 2 to get 70; 70 is still even, divide by 2 again to get 35; 35 is odd but ends in 5, so divide by 5 to get 7; and 7 is itself prime, so the tree stops there. Reading the divisors back gives 140 = 2 × 2 × 5 × 7 = 2² × 5 × 7 — matching the table above, and confirming that a composite number's factor tree always bottoms out in primes, however you branch it.

HCF and LCM, Read Straight Off the Factors

Once a number is broken into primes, HCF and LCM stop being separate calculations and become a matter of reading powers off the factorisation directly: HCF takes the smallest power of every prime common to all the numbers, and LCM takes the largest power of every prime that appears in any of them.

A quick way to check any two-number answer: for exactly two numbers, HCF × LCM always equals the product of the two numbers. For 72 and 108, that's 36 × 216 = 7776, and 72 × 108 = 7776 too — a match confirms both values are right. This shortcut only works for pairs, not for three or more numbers at once, which is why 12/15/21 and 17/23/29 above can't be checked the same way.

It's worth seeing why the smallest-power and largest-power rules work, rather than just applying them. A prime that appears in a number's factorisation to the power 3 can be pulled out of that number at most 3 times — so if a second number only has that same prime to the power 1, any factor shared by both numbers can use that prime at most once, which is exactly the smaller of the two powers. The LCM rule flips the logic: it needs to be divisible by both original numbers, so it must carry at least the largest power of every prime that shows up anywhere among them.

Can a Power of 6 Ever End in Zero?

Question 3 is a reasoning question rather than a calculation. A number ends in the digit 0 only if its prime factorisation contains both a 2 and a 5, since 10 = 2 × 5. Now write 6ⁿ in its own prime factorisation:

There is no factor of 5 anywhere in that expression, for any value of n. Since 5 never appears in the prime factorisation of 6ⁿ, it can never combine with a 2 to make a factor of 10 — so 6ⁿ cannot end in the digit 0, for any natural number n at all.

Every number that genuinely does end in 0 confirms the rule the other way round: 10 = 2 × 5, 20 = 2² × 5, and 100 = 2² × 5² all carry both primes. 6ⁿ, however it's expanded, can never pick up that missing 5 — no amount of multiplying 2s and 3s together will ever produce one.

Spotting a Composite Number Without a Calculator

Questions 4 and 5 test whether you can factor out a common term to reveal that an expression has more than two factors, which is the actual definition of a composite number — a number with a factor other than 1 and itself. You never need to compute the full value to prove this; finding just one such factor pair is enough.

• 7 × 11 × 13 + 13 — factor out 13: 13 × (7×11 + 1) = 13 × 78. The expression has 13 and 78 as factors, so it is composite.
• 7 × 6 × 5 × 4 × 3 × 2 × 1 + 5 — factor out 5: 5 × (7×6×4×3×2×1 + 1) = 5 × 1009. The expression has 5 and 1009 as factors, so it is composite.
• (17 × 11 × 2) + (17 × 11 × 5) — factor out 17 × 11: 17 × 11 × (2 + 5) = 17 × 11 × 7. Three separate prime factors confirm it is composite immediately.

A Pattern That Never Breaks: Powers of 6

The last question asks for the last digit of 6¹⁰⁰. Looking at the first few powers of 6 shows a pattern that locks in almost immediately:

• 6¹ = 6
• 6² = 36
• 6³ = 216
• 6⁴ = 1296

Every power of 6 ends in 6, because multiplying a number ending in 6 by 6 again always produces a result ending in 6 (6 × 6 = 36, and the pattern repeats no matter how large the exponent gets). So the last digit of 6¹⁰⁰ is 6, without needing to compute the full 78-digit number.

Continuing From Here

With the Fundamental Theorem in hand, the next natural step is Exercise 1.4's irrationality proofs, which lean on prime factorisation logic directly — the key supporting theorem there (if a prime p divides a², then p divides a) is really just another consequence of unique factorisation. You can also see HCF computed a different way, through repeated division rather than factor trees, back in Exercise 1.1, or revisit the number-system groundwork in the chapter introduction.