Math Problem Statement

What are th lowest common multiples?(a) 2 and 17 (b) 7 and 5 (c) 9 and 12 (d) 24f and 18f^2

Solution

The lowest common multiple (LCM) of two or more numbers is the smallest number that is a multiple of each of them.

Here are the calculations for the given pairs:


(a) 2 and 17

  • Prime factors of 2: 22
  • Prime factors of 17: 1717
  • Since 2 and 17 are coprime (no common factors), the LCM is their product: LCM(2,17)=2×17=34LCM(2, 17) = 2 \times 17 = 34

(b) 7 and 5

  • Prime factors of 7: 77
  • Prime factors of 5: 55
  • Since 7 and 5 are coprime, the LCM is their product: LCM(7,5)=7×5=35LCM(7, 5) = 7 \times 5 = 35

(c) 9 and 12

  • Prime factorization:
    • 9=329 = 3^2
    • 12=22×312 = 2^2 \times 3
  • The LCM takes the highest power of each prime factor:
    • 323^2 (from 9)
    • 222^2 (from 12)
  • LCM: LCM(9,12)=22×32=4×9=36LCM(9, 12) = 2^2 \times 3^2 = 4 \times 9 = 36

(d) 24f and 18f²

  • Prime factorization:
    • 24f=23×3×f24f = 2^3 \times 3 \times f
    • 18f2=2×32×f218f^2 = 2 \times 3^2 \times f^2
  • The LCM takes the highest power of each factor:
    • 232^3 (from 24f)
    • 323^2 (from 18f²)
    • f2f^2 (from 18f²)
  • LCM: LCM(24f,18f2)=23×32×f2=8×9×f2=72f2LCM(24f, 18f^2) = 2^3 \times 3^2 \times f^2 = 8 \times 9 \times f^2 = 72f^2

Final Results:

  • (a) LCM(2,17)=34LCM(2, 17) = 34
  • (b) LCM(7,5)=35LCM(7, 5) = 35
  • (c) LCM(9,12)=36LCM(9, 12) = 36
  • (d) LCM(24f,18f2)=72f2LCM(24f, 18f^2) = 72f^2

Do you want more detailed steps or have any questions? Here are 5 related questions to explore further:

  1. What is the difference between LCM and greatest common divisor (GCD)?
  2. How is the LCM used in solving equations with fractions?
  3. How can the LCM be calculated using the Euclidean algorithm?
  4. What are some real-life applications of LCM in scheduling problems?
  5. Can you find the LCM of more than two numbers simultaneously?

Tip: For polynomials, the LCM includes the highest powers of each variable and factor.

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Math Problem Analysis

Mathematical Concepts

Lowest Common Multiple (LCM)
Prime Factorization
Multiples

Formulas

LCM(a, b) = (a × b) / GCD(a, b)
LCM includes the highest powers of all prime factors involved

Theorems

Fundamental Theorem of Arithmetic (every integer > 1 can be uniquely factorized into prime numbers)

Suitable Grade Level

Grades 6-8