Kit Library / Mathematics / Number Theory
⚡ Topic Learning KitShared by a Veda learner · Generated with Veda AI
Veda Bites are swipeable micro-lessons — each one teaches exactly one idea. The bites below come from this kit — swipe through them in the app.
💡 Key Idea
One divides in, the other is divided by.
HCF and LCM are mirror concepts: HCF is the largest factor shared by the numbers, LCM is the smallest multiple shared by them.
↳ HCF is a factor of every number in the set; LCM is a multiple of every number in the set.
📖 Definition
Two words, one clean definition each.
↳ HCF = greatest shared divisor; LCM = least shared multiple.
➗ Formula
Multiply them and get the product.
↳ For two numbers, LCM = (a × b) ÷ HCF — a fast shortcut when one is known.
⚙️ Process
Break down, then pick min or max.
↳ HCF uses minimum exponents; LCM uses maximum exponents — same factorisation, opposite choice.
⚙️ Process
Divide, replace, repeat — HCF appears.
↳ Euclid's algorithm is the fastest hand method for HCF of two large numbers.
⚠️ Common Mistake
Same numbers, opposite question.
↳ Split into groups → HCF. Meet again / repeat together → LCM.
🌍 Real World
Tiles, bells, and bus timetables.
↳ If the answer must be *smaller than the inputs*, it's HCF; if *larger*, it's LCM.
🛠️ Application
Divide top and bottom by HCF.
A fraction is in lowest terms when its numerator and denominator share no common factor — that is, when you divide both by their HCF.
↳ HCF is the single divisor that reduces a fraction to its simplest form in one step.
🛠️ Application
LCM is the least common denominator.
To add or subtract fractions with different denominators, rewrite them using the LCM of the denominators as the common denominator.
↳ LCM keeps the common denominator as small as possible, so arithmetic stays simple.
No common factor? Instant answer.
If two numbers are co-prime (HCF = 1), their LCM is simply their product.
↳ Co-prime pair → LCM = product, HCF = 1.
🧠 Memory Trick
High factor, low multiple.
↳ The 'H' and 'L' in the names describe the *result*, not the exponent you pick.
🎯 Exam Tip
One known, the other found.
Exams frequently give you one of HCF or LCM plus the two numbers, and ask for the other. Use the identity directly instead of factorising.
↳ Whenever two of {HCF, LCM, product} are known, the third follows instantly.
🚫 Warning
The identity breaks with three numbers.
↳ The HCF × LCM = product rule is strictly a two-number tool.
❓ Did You Know
GCD and LCM — same thing.
↳ HCF, GCD, and GCF are interchangeable terms; LCD is LCM applied to denominators.
Two quick checks catch most errors.
↳ If HCF > smallest or LCM < largest, you've swapped them or made an arithmetic slip.
HCF (Highest Common Factor) and LCM (Lowest Common Multiple) are two fundamental number-theory tools. HCF finds the largest number that divides two or more numbers exactly, while LCM finds the smallest number that is a m…
Full explanation, worked examples & memory tricks in the app
All 17 questions, laddered from warm-up to topper-level, each with its answer and the reasoning behind it.
Q1. What is the HCF of $12$ and $18$?
BeginnerA · $6$
The largest number dividing both $12$ and $18$ is $6$ (since $12 = 6 \times 2$ and $18 = 6 \times 3$).
Q2. What is the LCM of $12$ and $18$?
BeginnerD · $36$
The smallest number that both $12$ and $18$ divide into is $36$.
Q3. Which statement correctly describes the LCM of two numbers?
BeginnerB · The smallest number that both divide into exactly
The LCM is the smallest number that is a multiple of both given numbers.
Q4. Two bells ring every $12$ minutes and every $18$ minutes. They ring together now. After how many minutes will they next ring together?
BeginnerB · $36$ minutes
The next simultaneous ring is the LCM of $12$ and $18$, which is $36$ minutes.
Q5. For any two positive integers $a$ and $b$, $\text{HCF}(a,b) \times \text{LCM}(a,b) = a \times b$.
BeginnerA · True
For any two positive integers $a$ and $b$, $\text{HCF}(a,b) \times \text{LCM}(a,b) = a \times b$.
Q6. For $24 = 2^3 \times 3$ and $36 = 2^2 \times 3^2$, what is the HCF?
IntermediateB · $2^2 \times 3 = 12$
HCF takes shared primes at their lowest powers: $2^2 \times 3 = 12$.
Q7. For $24 = 2^3 \times 3$ and $36 = 2^2 \times 3^2$, what is the LCM?
IntermediateD · $72$
LCM takes all primes at their highest powers: $2^3 \times 3^2 = 72$.
Q8. A student finds the HCF of $12$ and $18$ and gets $36$. Which sanity check proves this wrong?
IntermediateC · The HCF can never exceed the smaller number
The HCF is a divisor, so it cannot be larger than the smaller number. $36$ cannot divide $12$.
Q9. If $\text{HCF}(a,b) = 4$ and $a \times b = 96$, what is $\text{LCM}(a,b)$?
IntermediateA · $24$
Using $\text{HCF} \times \text{LCM} = a \times b$: $4 \times \text{LCM} = 96$, so $\text{LCM} = 24$.
Q10. Which pair of numbers is coprime?
IntermediateB · $8$ and $15$
$8 = 2^3$ and $15 = 3 \times 5$ share no prime factor, so their HCF is $1$ — they are coprime.
Q11. To simplify $\frac{18}{24}$ in one step, which value should you divide numerator and denominator by?
IntermediateB · The HCF of $18$ and $24$
Dividing by the HCF ($6$) gives $\frac{3}{4}$ directly in one step.
Q12. A carpenter has boards of length $48$ cm and $18$ cm and wants the largest identical planks that fit both exactly. Which tool does he need?
IntermediateB · The HCF of $48$ and $18$
Cutting the largest identical pieces is a splitting problem, so the HCF ($6$ cm) is the answer.
Q13. For $4$, $6$ and $8$, the HCF is $2$ and the LCM is $24$. Which statement is true?
AdvancedB · $\text{HCF} \times \text{LCM} \ne 4 \times 6 \times 8$ because the product rule needs exactly two numbers
$2 \times 24 = 48$ but $4 \times 6 \times 8 = 192$. The product rule applies only to two numbers.
Q14. Two gears have $24$ and $36$ teeth. A marked tooth on each is aligned. After how many tooth-passes will the marked teeth next align?
AdvancedB · $72$ tooth-passes
Alignment recurs at the LCM of $24$ and $36$, which is $72$.
Q15. Applying the Euclidean algorithm to $48$ and $18$: $48 = 18 \times 2 + 12$, then $18 = 12 \times 1 + 6$, then $12 = 6 \times 2 + 0$. What is the HCF?
AdvancedD · $6$
The last non-zero remainder is $6$, so $\text{HCF}(48, 18) = 6$.
Q16. For two positive integers, which statement best explains why a large HCF forces a small LCM?
AdvancedB · Because their product is fixed at $a \times b$, so one must shrink as the other grows
The product rule $\text{HCF} \times \text{LCM} = a \times b$ fixes the product, so the two values trade off.
Q17. Given two integers, $a = 24$ and $b = 36$, calculate the HCF and LCM. Then, if the product of the two integers is $a \times b$, verify the product rule. What is the value of $\text{LCM}(a,b)$?
AdvancedA · 72
The HCF of $24$ and $36$ is $12$, and the LCM is calculated as $\text{LCM}(a,b) = \frac{a \times b}{\text{HCF}(a,b)} = \frac{24 \times 36}{12} = 72$. Thus, the correct value of $\text{LCM}(24,36)$ is $72$.
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