LEAST COMMON MULTIPLE
Excerpts
LEAST COMMON MULTIPLE
To find the L.C.M. of two or more numbers, find the prime factors of each number, take the product of the different factors, using each the greatest number of times it is found in any of the numbers
LEAST COMMON MULTIPLE
$12 a^{2} b^{3}$ is the least common multiple of $3 a^{2} b$ and $4 a b^{3}$. How many of the factors of $3 a^{2} b$ are found in the L.C.M.?
LEAST COMMON MULTIPLE
This exercise should be conducted orally, and if, at any point, the students do not readily recall the method, the examples of that class should be duplicated till the principle is clear.
Equations
Problems
Exercise 40
Exercise 40, problem 1
$a^{2} - 4$ and $a^{2} + 3 a - 10$.
Printed answer:- $\left(a^2-4\right)\left(a+5\right)$.
verified: the printed answer passed a computed check
How it was checked
lcm: passes(a**2 - 4)*(a + 5)
Exercise 40, problem 10
$x^6 + x^3$, $2x^4 - 2x^2$, and $x^2 + x$.
Printed answer:- $2x^3\left(x^3+1\right)\left(x-1\right)$.
verified: the printed answer passed a computed check
How it was checked
lcm: passes2*x**3*(x**3 + 1)*(x - 1)
Exercise 40, problem 11
$a^4 + 2a^2 + 1$, $1 - 2a^2 + a^4$, and $1 - a^4$.
Printed answer:- $\left(1-a^4\right)^2$.
verified: the printed answer passed a computed check
How it was checked
lcm: passes(1 - a**4)**2
Exercise 40, problem 12
$a^2z - x^2z$, $3ax - 3x^2$, and $2ax + 2a^2$.
Printed answer:- $6axz\left(a^2 - x^2\right)$.
verified: the printed answer passed a computed check
How it was checked
lcm: passes6*a*x*z*(a**2 - x**2)
Exercise 40, problem 13
$am + an - 3m - 3n$, $m^2 - n^2$, and $a^2 - 10a + 21$.
Printed answer:- $\left(m^2-n^2\right)\left(a^2-10a+21\right)$.
verified: the printed answer passed a computed check
How it was checked
lcm: passes(m**2 - n**2)*(a**2 - 10*a + 21)
Exercise 40, problem 14
$3b - 3b^2$, $1 - b^3$, and $2x - 2b^2x$.
Printed answer:- $6bx\left(1-b^3\right)\left(1+b\right)$.
verified: the printed answer passed a computed check
How it was checked
lcm: passes6*b*x*(1 - b**3)*(1 + b)
Exercise 40, problem 15
$a^2 - x^2 - a - x$, $5a^2b - 10abx + 5bx^2$, and $a^2b - abx - ab$.
Printed answer:- $5ab\left(a+x\right)\left(a-x\right)^2\left(a-x-1\right)$.
verified: the printed answer passed a computed check
How it was checked
lcm: passes5*a*b*(a + x)*(a - x)**2*(a - x - 1)
Exercise 40, problem 16
The thermometer now indicates $c$ degrees above zero, but yesterday it indicated $y$ degrees below zero. How much warmer is it to-day than yesterday?
Printed answer:- $c + y$ degrees.
verified: the printed answer passed a computed check
How it was checked
identity: passes, with the problem read into an equationc + y
Exercise 40, problem 17
A certain county extends from $b$ degrees north latitude to $y$ degrees north latitude. What is the extent of the county from north to south?
Printed answer:- $b - y$, or $y - b$ degrees.
verified: the printed answer passed a computed check
How it was checked
identity: passes, with the problem read into an equationb - y
Exercise 40, problem 18
If A can build a wall in $x$ days, what part of the wall will he build in two days?
Printed answer:- $\frac{2}{x}$.
verified: the printed answer passed a computed check
How it was checked
identity: passes, with the problem read into an equation2/x
Exercise 40, problem 19
How many times can $x^3 + 12$ be subtracted from $x^6 + 24 x^3 + 144$?
Printed answer:- $x^3 + 12$.
verified: the printed answer passed a computed check
How it was checked
identity: passes, with the problem read into an equationx**3 + 12
Exercise 40, problem 2
$x^{3} + 1$ and $x^{2} + 2 x + 1$.
Printed answer:- $\left(x^3+1\right)\left(x+1\right)$.
verified: the printed answer passed a computed check
How it was checked
lcm: passes(x**3 + 1)*(x + 1)
Exercise 40, problem 20
Divide $2142 between two men so that one shall receive six times as much as the other.
Printed answer:- $ 306; $ 1836.
verified: the printed answer passed a computed check
How it was checked
solve: passes, with the problem read into an equation{s: 306, L: 1836}
Exercise 40, problem 3
$x^{2} - 2 x - 15$ and $x^{2} - 9$.
Printed answer:- $\left(x^2-9\right)\left(x-5\right)$.
verified: the printed answer passed a computed check
How it was checked
lcm: passes(x**2 - 9)*(x - 5)
Exercise 40, problem 4
$x^{4} - 4 x^{2} + 3$ and $x^{6} - 1$.
Printed answer:- $\left(x^6-1\right)\left(x^2-3\right)$.
verified: the printed answer passed a computed check
How it was checked
lcm: passes(x**6 - 1)*(x**2 - 3)
Exercise 40, problem 5
$x^2 - 1$, $x^2 - 4x + 3$, and $2x^2 - 2x - 12$.
Printed answer:- $2\left(x^2-1\right)\left(x-3\right)\left(x+2\right)$.
verified: the printed answer passed a computed check
How it was checked
lcm: passes2*(x**2 - 1)*(x - 3)*(x + 2)
Exercise 40, problem 6
$a^3 + 1 + 3a + 3a^2$ and $am - 2 - 2a + m$.
Printed answer:- $\left(a+1\right)^3\left(m-2\right)$.
verified: the printed answer passed a computed check
How it was checked
lcm: passes(a + 1)**3*(m - 2)
Exercise 40, problem 7
$a^2 - 4$, $a^2 + 3a - 10$, $a^2 - 25$, $a^2 - 9$, $a^2 - 8a + 15$, and $a^2 + 5a + 6$.
Printed answer:- $\left(a^2-4\right)\left(a^2-25\right)\left(a^2-9\right)$.
verified: the printed answer passed a computed check
How it was checked
lcm: passes(a**2 - 4)*(a**2 - 25)*(a**2 - 9)
Exercise 40, problem 8
$x^3 - 1 - 3x^2 + 3x$ and $xy - 3 - y +3x$.
Printed answer:- $\left(x-1\right)^3\left(y+3\right)$.
verified: the printed answer passed a computed check
How it was checked
lcm: passes(x - 1)**3*(y + 3)
Exercise 40, problem 9
$x^2 - 1$, $x^2 - 9$, $x^2 - 2x - 3$, $x^2 - 16$, $x^2 - x - 12$, $x^2 + 5x + 4$.
Printed answer:- $\left(x^2-1\right)\left(x^2-9\right)\left(x^2-16\right)$.
verified: the printed answer passed a computed check
How it was checked
lcm: passes(x**2 - 1)*(x**2 - 9)*(x**2 - 16)
Exercise 41
The data holds no problems for this exercise yet.