lcm (205,797,003,129; 627,002,221,049) = ?
Method 1. The prime factorization:
The prime factorization of a number: finding the prime numbers that multiply together to make that number.
205,797,003,129 = 33 × 251 × 30,366,977
205,797,003,129 is not a prime number but a composite one.
627,002,221,049 = 677 × 10,711 × 86,467
627,002,221,049 is not a prime number but a composite one.
- Prime number: a natural number that is only divisible by 1 and itself. A prime number has exactly two factors: 1 and itself.
- Composite number: a natural number that has at least one other factor than 1 and itself.
Calculate the least common multiple, lcm:
Multiply all the prime factors of the two numbers. If there are common prime factors then only the ones with the largest exponents are taken (the largest powers).
The least common multiple:
lcm (205,797,003,129; 627,002,221,049) = 33 × 251 × 677 × 10,711 × 86,467 × 30,366,977 = 129,035,178,047,111,002,662,321
The two numbers have no prime factors in common
129,035,178,047,111,002,662,321 = 205,797,003,129 × 627,002,221,049
Method 2. The Euclidean Algorithm:
1. Calculate the greatest (highest) common factor (divisor):
- This algorithm involves the process of dividing numbers and calculating the remainders.
- 'a' and 'b' are the two natural numbers, 'a' >= 'b'.
- Divide 'a' by 'b' and get the remainder of the operation, 'r'.
- If 'r' = 0, STOP. 'b' = the gcf (hcf, gcd) of 'a' and 'b'.
- Else: Replace ('a' by 'b') and ('b' by 'r'). Return to the step above.
Step 1. Divide the larger number by the smaller one:
627,002,221,049 ÷ 205,797,003,129 = 3 + 9,611,211,662
Step 2. Divide the smaller number by the above operation's remainder:
205,797,003,129 ÷ 9,611,211,662 = 21 + 3,961,558,227
Step 3. Divide the remainder of the step 1 by the remainder of the step 2:
9,611,211,662 ÷ 3,961,558,227 = 2 + 1,688,095,208
Step 4. Divide the remainder of the step 2 by the remainder of the step 3:
3,961,558,227 ÷ 1,688,095,208 = 2 + 585,367,811
Step 5. Divide the remainder of the step 3 by the remainder of the step 4:
1,688,095,208 ÷ 585,367,811 = 2 + 517,359,586
Step 6. Divide the remainder of the step 4 by the remainder of the step 5:
585,367,811 ÷ 517,359,586 = 1 + 68,008,225
Step 7. Divide the remainder of the step 5 by the remainder of the step 6:
517,359,586 ÷ 68,008,225 = 7 + 41,302,011
Step 8. Divide the remainder of the step 6 by the remainder of the step 7:
68,008,225 ÷ 41,302,011 = 1 + 26,706,214
Step 9. Divide the remainder of the step 7 by the remainder of the step 8:
41,302,011 ÷ 26,706,214 = 1 + 14,595,797
Step 10. Divide the remainder of the step 8 by the remainder of the step 9:
26,706,214 ÷ 14,595,797 = 1 + 12,110,417
Step 11. Divide the remainder of the step 9 by the remainder of the step 10:
14,595,797 ÷ 12,110,417 = 1 + 2,485,380
Step 12. Divide the remainder of the step 10 by the remainder of the step 11:
12,110,417 ÷ 2,485,380 = 4 + 2,168,897
Step 13. Divide the remainder of the step 11 by the remainder of the step 12:
2,485,380 ÷ 2,168,897 = 1 + 316,483
Step 14. Divide the remainder of the step 12 by the remainder of the step 13:
2,168,897 ÷ 316,483 = 6 + 269,999
Step 15. Divide the remainder of the step 13 by the remainder of the step 14:
316,483 ÷ 269,999 = 1 + 46,484
Step 16. Divide the remainder of the step 14 by the remainder of the step 15:
269,999 ÷ 46,484 = 5 + 37,579
Step 17. Divide the remainder of the step 15 by the remainder of the step 16:
46,484 ÷ 37,579 = 1 + 8,905
Step 18. Divide the remainder of the step 16 by the remainder of the step 17:
37,579 ÷ 8,905 = 4 + 1,959
Step 19. Divide the remainder of the step 17 by the remainder of the step 18:
8,905 ÷ 1,959 = 4 + 1,069
Step 20. Divide the remainder of the step 18 by the remainder of the step 19:
1,959 ÷ 1,069 = 1 + 890
Step 21. Divide the remainder of the step 19 by the remainder of the step 20:
1,069 ÷ 890 = 1 + 179
Step 22. Divide the remainder of the step 20 by the remainder of the step 21:
890 ÷ 179 = 4 + 174
Step 23. Divide the remainder of the step 21 by the remainder of the step 22:
179 ÷ 174 = 1 + 5
Step 24. Divide the remainder of the step 22 by the remainder of the step 23:
174 ÷ 5 = 34 + 4
Step 25. Divide the remainder of the step 23 by the remainder of the step 24:
5 ÷ 4 = 1 + 1
Step 26. Divide the remainder of the step 24 by the remainder of the step 25:
4 ÷ 1 = 4 + 0
At this step, the remainder is zero, so we stop:
1 is the number we were looking for - the last non-zero remainder.
This is the greatest (highest) common factor (divisor).
The greatest (highest) common factor (divisor):
gcf, hcf, gcd (205,797,003,129; 627,002,221,049) = 1
2. Calculate the least common multiple:
The least common multiple, Formula:
lcm (a; b) = (a × b) / gcf, hcf, gcd (a; b)
lcm (205,797,003,129; 627,002,221,049) =
(205,797,003,129 × 627,002,221,049) / gcf, hcf, gcd (205,797,003,129; 627,002,221,049) =
129,035,178,047,111,002,662,321 / 1 =
129,035,178,047,111,002,662,321
The least common multiple:
lcm (205,797,003,129; 627,002,221,049) = 129,035,178,047,111,002,662,321 = 33 × 251 × 677 × 10,711 × 86,467 × 30,366,977
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More multiples starting from the least common multiple
- Any common multiple of two numbers is also a multiple of the least common multiple, LCM, of twose two numbers.
The following numbers are also multiples of 205,797,003,129 and 627,002,221,049:
129,035,178,047,111,002,662,321 × 0 = 0
129,035,178,047,111,002,662,321 × 2 = 258,070,356,094,222,005,324,642
129,035,178,047,111,002,662,321 × 3 = 387,105,534,141,333,007,986,963
...
- There are infinitely many multiples of any two numbers.
How to check if a number is a common multiple of two numbers?
- After calculating the LCM, divide the number to be checked by the LCM. If the remainder of this division is zero, then the number being checked is a multiple of the other two numbers. If the remainder is not zero, then the number being checked is not a multiple.
- For example: the LCM of the numbers 4 and 6 is 2 × 2 × 3 = 12.
- Question: is 36 a multiple of the numbers 4 and 6? Answer: 36 ÷ 12 = 3 and the remainder is 0, so 36 is a multiple of 4 and 6.
- Question: is 28 a multiple of the numbers 4 and 6? Answer: 28 ÷ 12 = 2 and the remainder is 4, so 28 is not a multiple of 4 and 6.
Why is it useful to calculate the least common multiple?
- In order to add, subtract or sort fractions with different denominators, we must make their denominators the same. An easy way is to calculate the least common multiple of all the denominators (the least common denominator).
- By definition, the least common multiple of two numbers is the smallest natural number that is: (1) greater than 0 and (2) a multiple of both numbers.