HCF and LCM Concepts

1. HCF–LCM Relationship

• Formula: $\boxed{HCF(a,b)\times LCM(a,b)=a\times b}$
• Example: Product of two numbers is 960 and their HCF is 8. Find the LCM. Solution: $LCM=\frac{960}{8}=120$

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2. Finding One Number Using HCF and LCM

• Formula: If one number, HCF and LCM are given: $\boxed{\text{Other number}=\frac{HCF\times LCM}{\text{Given number}}}$
• Example: HCF of two numbers is 6 and LCM is 180. If one number is 30, find the other. Solution: $\text{Other number}=\frac{6\times180}{30}=36$

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3. HCF and LCM by Prime Factorization

• Formula: For $a=p_1^{a_1}p_2^{a_2}\cdots$ $b=p_1^{b_1}p_2^{b_2}\cdots$ $\boxed{HCF=\prod p^{\min(\text{powers})}}$ $\boxed{LCM=\prod p^{\max(\text{powers})}}$
• Example: Find HCF and LCM of 24 and 36. Solution: $24=2^3\times3,\qquad36=2^2\times3^2$ $HCF=2^2\times3=12$ $LCM=2^3\times3^2=72$

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4. HCF and LCM of Three or More Numbers

• Formula: HCF → take the minimum powers of common primes. LCM → take the maximum powers of all primes.
• Example: Find HCF and LCM of 24, 36 and 48. Solution: $24=2^3\times3$ $36=2^2\times3^2$ $48=2^4\times3$ $HCF=2^2\times3=12$ $LCM=2^4\times3^2=144$

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5. Greatest Number Leaving the Same Remainder

• Formula: If a number leaves the same remainder when divided by several numbers: $\boxed{\text{Required divisor}=HCF(a-r,b-r,c-r,\ldots)}$
• Example: Find the greatest number that divides 867 and 255 leaving remainder 3 in each case. Solution: $867-3=864,\qquad255-3=252$ $HCF(864,252)=36$ Answer: 36

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6. Least Number Leaving the Same Remainder

• Formula: If the required number leaves remainder (r) when divided by (a,b,c): $\boxed{N=LCM(a,b,c)+r}$
• Example: Find the least number which leaves remainder 7 when divided by 12 and 18. Solution: $LCM(12,18)=36$ $N=36+7=43$

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7. Least Number Divisible by Given Numbers

• Formula: $\boxed{N=LCM(a,b,c,\ldots)}$
• Example: Find the smallest number divisible by 15, 20 and 30. Solution: $LCM(15,20,30)=60$ Answer: 60

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8. HCF-Based Equal Grouping

• Formula: $\boxed{\text{Largest possible group size}=HCF(\text{given quantities})}$
• Example: 96, 144 and 240 books are to be divided into equal groups of maximum size. Find the group size. Solution: $HCF(96,144,240)=48$ Answer: 48 books per group.

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9. LCM-Based Repeating Events

• Formula: $\boxed{\text{Next simultaneous occurrence}=LCM(\text{intervals})}$
• Example: Three bells ring every 12, 18 and 24 minutes. If they ring together at 9:00 AM, when will they next ring together? Solution: $LCM(12,18,24)=72\text{ min}$ Answer: 10:12 AM

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10. HCF + LCM in Word Problems

• Formula: Use: $HCF\times LCM=a\times b$ together with the given condition.
• Example: Two numbers have HCF 12 and LCM 180. If one number is 36, find the other. Solution: $12\times180=36\times x$ $x=60$ Answer: 60

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11. HCF of Fractions

• Formula: For fractions in lowest form: $\boxed{HCF=\frac{HCF(\text{numerators})}{LCM(\text{denominators})}}$
• Example: Find the HCF of (\frac5{12}) and (\frac7{18}). Solution: $HCF(5,7)=1,\qquad LCM(12,18)=36$ $HCF=\frac1{36}$

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12. LCM of Fractions

• Formula: For fractions in lowest form: $\boxed{LCM=\frac{LCM(\text{numerators})}{HCF(\text{denominators})}}$
• Example: Find the LCM of (\frac34,\frac56,\frac78). Solution: $LCM(3,5,7)=105$ $HCF(4,6,8)=2$ $LCM=\frac{105}{2}$

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Advanced HCF–LCM Variants

13. Number Pair from HCF and LCM

If $HCF=h,\qquad LCM=l$ then write the numbers as $a=hx,\qquad b=hy$ where $\boxed{xy=\frac lh}$ and (x,y) are coprime.

• Example: HCF = 6 and LCM = 180. Find possible pairs. Solution: $xy=\frac{180}{6}=30$ Coprime factor pairs of 30: $(1,30),(2,15),(3,10),(5,6)$ Therefore possible pairs: $(6,180),(12,90),(18,60),(30,36)$

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14. HCF and LCM with a Given Difference

• Formula: If two numbers are (hx) and (hy): $\boxed{h(y-x)=\text{difference}}$ along with $\boxed{xy=\frac{LCM}{HCF}}$

• Example: HCF of two numbers is 6, LCM is 180, and their difference is 24. Find the numbers. Solution: $xy=30$ $6(y-x)=24\Rightarrow y-x=4$ Coprime factors satisfying this are (x=5,y=9): $5\times9=45\neq30$ So no such pair exists.

This type is useful because the conditions must be checked for consistency.

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15. HCF–LCM with Three Numbers

For three numbers, do not use $HCF\times LCM=a\times b\times c$ as a general rule. That identity is only directly valid for two numbers.

For three numbers, use prime factorization: $\boxed{HCF=\prod p^{\min(e_1,e_2,e_3)}}$ $\boxed{LCM=\prod p^{\max(e_1,e_2,e_3)}}$