Answer:
GCD (or HCF) and LCM of 3 numbers (with formula and without)
Formula:
LCM (p,q,r) = [ GCD (p,q,r) × LCM (p,q) × LCM (p,r) × LCM (q,r) ] ÷ p × q × r
I don't know if I'm the first one to discover (probably not), but this is the formula that I've discovered right now about the relationship between the GCD and LCM of 3 numbers.
There's a much simpler way to find the LCM and GCD of 3 numbers, but it doesn't include any relationship formula.
To find GCD, do factorization with prime numbers. It's very simple.
To find LCM you do the following:
1) Factorize each one of the numbers separately.
2) Find the largest number of times a same prime number appears with each one of the prime numbers in each one of the separated factorization.
3) Multiply all the prime numbers by the number of times they appear, then multiply all of the products together.
Example: LCM (12,42,90)
Prime factors of 12 -> 2, 2, 3
Prime factors of 42 -> 2, 3, 7
Prime factors of 90 -> 2, 3, 3, 5
Largest number of times that each prime number appears:
2 -> 2 times (in the factorization of 12)
3 -> 2 times (in the factorization of 90)
5 -> 1 time
7 -> 1 time
2² = 4
3² = 9
5¹ = 5
7¹ = 7
4 × 9 × 5 × 7 = 1260
LCM (12,42,90) = 1260
Obtaining the LCM with the formula:
LCM (12,42,90) = [ GCD (12,42,90) × LCM (12,42) × LCM (12,90) × LCM (42,90) ] ÷ 12 × 42 × 90
LCM (12,42,90) = [ 6 × 84 × 180 × 630 ] ÷ 45360
LCM (12,42,90) = 1260
