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Prime Factors
Every number is either a prime number or a composite number. You cannot evenly divide a prime number by any number but itself and one. A composite number is a product of a finite number of prime factors. Seven is a prime number. But six is a composite number, a product of the prime numbers two and three.
No Prime Factors in Common
Suppose you were asked to add 1/7 and 1/6. There are no common prime factors between the denominators. Seven is a prime number itself, and as such is its own only prime factor. Six is the product of the prime numbers two and three. Whenever there are no prime factors between the two denominators, the lowest common denominator is their product -- in this case, 42. That's the lowest number that both six and seven can divide evenly into.
Prime Factors in Common
Suppose you were asked to add the fractions 1/6 and 1/14. Six has the prime factors two and three; 14 has the prime factors two and seven. The product of six and fourteen is 84, so obviously 84 will divide by each number evenly. But the lowest common denominator between 6 and 14 is 42. If the two denominators have a prime factor in common, the lowest common denominator is less than the product of the two denominators.
Determining Lowest Common Denominator
To find the lowest common denominator between two numbers, write down all prime factors or both. If a prime factor is repeated -- a factor of both denominators -- cross out one of them. Now multiply all of the numbers that remain. The result is the lowest common denominator. If you were adding 1/6 and 1/14, you would write out two and three for the factors of six, and two and seven for the factors of fourteen. Crossing out one of the twos, you're left with two, three and seven, the product of which is 42.
Repeated Prime Factors
If a prime factor is repeated, list it twice when determining the lowest common denominator. Suppose you wanted to add 1/70 and 1/50. 70 has the prime factors two, five and seven. Fifty has the prime factors two, five and five. Cross out one of the twos, because it's a factor of both. Five is also a factor of both, so cross out one of them. You are left with seven, two, five, and five, the product of which is 350. If you wanted to find the greatest common denominator between 1/50 and 1/75, write out five, five and two for the factors of fifty, and five, five and three for the factors of 75. Both have two factors of five, so cross out two of them, leaving five, five, three and two as your remaining factors. Their product is 150, which is the lowest common denominator.
Adding Fractions
Once you have determined the lowest common denominator, adding fractions is easy. If you wanted to add 1/6 and 1/14, you know that the greatest common denominator is 42. 1/6 is 7/42. 1/14 is 3/42. The sum is 10/42. This can be reduced to 5/21.