02 // Visual Math & Finance SolversGCD(A,B) × LCM(A,B) = A × B

Visual LCM & GCD (HCF) Prime Factor Venn & Tree Solver

Given prime factorizations $A = \prod p_i^{a_i}$ and $B = \prod p_i^{b_i}$, the Greatest Common Divisor takes minimum exponents $\gcd(A,B) = \prod p_i^{\min(a_i,b_i)}$, the Least Common Multiple takes maximum exponents $\text{lcm}(A,B) = \prod p_i^{\max(a_i,b_i)}$, and $\gcd(A,B) \times \text{lcm}(A,B) = A \times B$.

GCD(A,B) × LCM(A,B) = A × B

Inputs & Parameters (2)

First Integer (A)

2±1720

Second Integer (B)

2±1720

Prime Factorization (84)

2^2 × 3 × 7

4 prime leaves in tree

Total Divisors τ(84)

12 divisors

Smallest prime factor = 2

GCD(84, 126) [HCF]

Shared factors: 2×3×7

LCM(84, 126)

252

Check: 42 × 252 = 10,584

20-Row Comparison Table

Compare across:

#	First Integer (A)	Prime Factor Count Ω(N)	LCM(N, M)	Prime Factors of N
01	2	2	126	2
02	40	2	2,520	2 × 2 × 2 × 5
03	78	6	1,638	2 × 3 × 13
04	115	1	14,490	5 × 23
05	153	9	2,142	3 × 3 × 17
06	191	1	24,066	191
07	229	1	28,854	229
08	267	3	11,214	3 × 89
09	304	2	19,152	2 × 2 × 2 × 2 × 19
10	342	18	2,394	2 × 3 × 3 × 19
11	380	2	23,940	2 × 2 × 5 × 19
12	418	2	26,334	2 × 11 × 19
13	455	7	8,190	5 × 7 × 13
14	493	1	62,118	17 × 29
15	531	9	7,434	3 × 3 × 59
16	569	1	71,694	569
17	607	1	76,482	607
18	644	14	5,796	2 × 2 × 7 × 23
19	682	2	42,966	2 × 11 × 31
20	720	18	5,040	2 × 2 × 2 × 2 × 3 × 3 × 5

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Key Analytical Takeaways

01The shared prime factors in the overlap of $A$ and $B$ multiply to form $\gcd(A,B)$ (also called Highest Common Factor, HCF).
02Multiplying the shared $\gcd$ primes by the remaining unshared primes of both numbers yields $\text{lcm}(A,B)$.
03When $\gcd(A,B) = 1$, the two integers share no prime factors and are called coprime (relatively prime), so $\text{lcm}(A,B) = A \times B$.

Frequently Asked Questions

What is the difference between GCD, GCF, and HCF?+

Greatest Common Divisor (GCD), Greatest Common Factor (GCF), and Highest Common Factor (HCF) are three names for the exact same value: the largest positive integer that divides both A and B with zero remainder.

How is LCM used when adding fractions?+

The Least Common Denominator (LCD) of two fractions a/b and c/d is simply the LCM of their denominators b and d.

What does it mean when GCD(A, B) equals 1?+

When two integers share no common prime factors, their GCD is 1; they are called coprime (or relatively prime), and their least common multiple is simply their product A × B.