Answer to Question #92943 in Algebra for kingmosha

Question

Find the gcf and lcm of the following:
(a) 3m^2pq,9mp^2,12mpq,3mn^2

(b) 5a^2b^3,10ab^4,2a^2b^3

(c) (m - n)^2, (m - n)

Answer 1

(a) Find the GCF and LCM of these sets of expressions: 3m²pq, 9mp², 12mpq, 3mn².

Solution: List the prime factors of each.

3m²pq : 3*m*m*p*q

9mp² : 3*3*m*p*p

12mpq: 3*2*2*m*p*q

3mn²: 3*m*n*n

3*m is the only common factor; therefore, 3*m is the GCF.

Multiply each factor the greatest number of times it occurs in any of the numbers.

3m²pq has two m's and one q,

9mp² has two 3's and two p's,

12mpq has two 2's,

3mn² has two n's,

so we multiply m two times, q once, 3 two times, p two times, 2 two times, n two times.

This gives 64m²qp²n² is LCM, the smallest expression that can be divided evenly by 3m²pq, 9mp², 12mpq, 3mn².

(b) Find the GCF and LCM of these sets of numbers: 5a²b³,10ab⁴, 2a²b³.

Solution: List the prime factors of each.

5a²b³: 5*a*a*b*b*b

10ab⁴: 2*5*a*b*b*b*b

2a²b³: 2*a*a*b*b*b

a*b³ is the only common factor; therefore, a*b³ is the GCF.

Multiply each factor the greatest number of times it occurs in any of the numbers.

5a²b³: has one 5 and two a's,

10ab⁴ has one 2 and four b's,

so we multiply 2 once, 5 two times, a two times, b four times.

This gives 10a²b⁴ is LCM, the smallest expression that can be divided evenly by 5a²b³,10ab⁴, 2a²b³.

(c) Find the GCF and LCM of these sets of numbers: (m-n)², (m-n).

Solution: List the prime factors of each.

(m-n)² : (m-n)*(m-n)

(m-n) : (m-n)

(m-n) is the only common factor; therefore, (m-n) is the GCF.

Multiply each factor the greatest number of times it occurs in any of the numbers. (m-n)² has two (m-n), and (m-n) has one (m-n), so we multiply (m-n) two times. This gives (m-n)² is LCM, the smallest expression that can be divided evenly by (m-n)² and (m-n).