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Q: X is at least 2

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Factorise into prime numbers and collect the least set of prime numbers: 8 = 2 x 2 x 2 30 = 5 x 3 x 2 Least set is 2 x 2 x 2 x 5 x 3 (miss out the 2 - it appears in both 30 primes and 8 primes) Least common multiple is 2 x 2 x 2 x 5 x 3 = 120

40 = 2 x 2 x 2 x 5 30 = 2 x 3 x 5 Least common denominator = 2 x 2 x 2 x 5 x 3 = 120

2 x 2 x 2 x 2 x 3

2 x 2 x 2 x 2 x 2 x 5 = 160

12 = 2 x 2 x 3; 40 = 2 x 2 x 2 x 5; 52 = 2 x 2 x 13. The HIGHEST Common Factor is 4; The Least Common MULTIPLE is 1560.

2 x 2 x 2 x 3 x 3 x 5

240 = 2 x 2 x 2 x 2 x 3 x 5

2 x 2 x 2 x 3 x 5 = 180

14 = 2 x 7 16 = 2 x 2 x 2 x 2 ---------------------- The LCM of 14,16 is: 2 x 2 x 2 x 2 x 7 = 112

Find the prime factorization of each: 24 = 2 x 2 x 2 x 3 30 = 2 x 3 x 5 The least common multiple is therefore 2 x 2 x 2 x 3 x 5, or 120.

Least common multiple = 7x X 2 = 14x

2 x 2 x 3 x 3 x 5

2 x 2 x 2 x 3 x 7 = 168

10 = 10 x 1 40 = 10 x 2 x 2 100 = 10 x 2 x 5 least common multiple = 10 x 2 x 2 x 5 = 200

1

84 = 2 x 2 x 3 x 7, and 721 = 7 x 103. Therefore the least common multiple of 721 and 84 is 8,652.

2 x 2 x 2 x 2 = 16 2 x 2 x 2 x 3 x 3 = 72 2 x 2 x 2 x 2 x 3 x 3 = 144, the LCM

The least common multiple is 24. 12 x 2 = 24, 8 x 3 = 24 and 2 x 12 = 24.

To find the least common multiple of 2, 3, and 4, factor them: 2 = 2 x 1 3 = 3 x 1 4 = 2 x 2 (Since one 2 has already been used to factor the number 2, only one 2 counts in the factors for the number 4.) The least common multiple of 2, 3, and 4 = 1 x 2 x 3 x 2 = 12.

The lowest (or least) common multiple of 56 and 54 is 1,512 This can be found by collecting the least set of prime number factors of both 54 and 56. The prime factors of 54 are (54 = 27 x 2 = 9 x 3 x 2 =) 3 x 3 x 3 x 2. The prime number factors of 56 are (56 = 28 x 2 = 14 x 2 x 2 =) 7 x 2 x 2 x 2 . The lowest common multiple can now be found by looking at both sets of prime number factors. 2 appears in both sets so we do not need to repeat in the answer, all other numbers are multiplied. The LCM is then 7 x 2 x 2 x 2 x 3 x 3 x 3 = 1,512

2 x 2 x 2 = 8 2 x 3 x 3 = 18 2 x 2 x 2 x 3 x 3 = 72, the LCM

6

Factor them. 2 x 3 x 3 = 18 2 x 2 x 2 x 2 = 16 2 x 2 x 2 x 2 x 3 x 3 = 144, the LCM

2 x 2 x 2 x 2 = 16 2 x 2 x 2 x 3 = 24 2 x 2 x 2 x 2 x 3 = 48, the LCM

57 = 3 x 19, 76 = 2 x 2 x 19 so LCM = 3 x 19 x 2 x 2 = 228