<span>We convert the sentences to mathematical expressions as follows
"</span>Four times the sum of the least and greatest integers" : 4[n+(n+2)]
"is 12 less than three times the least integer": = 3n-12
So we have:
4[n+(n+2)]=3n-12 (this is the equation)
4[2n+2]=3n-12
8n+8=3n-12
8n-3n=-12-8
5n=-20
n=-20/5=-4
Answer: the equation is 4[n+(n+2)]=3n-12 , the least integer is -4
Solve for x:
10 (x + 2) = 5 (x + 8)
Expand out terms of the left hand side:
10 x + 20 = 5 (x + 8)
Expand out terms of the right hand side:
10 x + 20 = 5 x + 40
Subtract 5 x from both sides:
(10 x - 5 x) + 20 = (5 x - 5 x) + 40
10 x - 5 x = 5 x:
5 x + 20 = (5 x - 5 x) + 40
5 x - 5 x = 0:
5 x + 20 = 40
Subtract 20 from both sides:
5 x + (20 - 20) = 40 - 20
20 - 20 = 0:
5 x = 40 - 20
40 - 20 = 20:
5 x = 20
Divide both sides of 5 x = 20 by 5:
(5 x)/5 = 20/5
5/5 = 1:
x = 20/5
The gcd of 20 and 5 is 5, so 20/5 = (5×4)/(5×1) = 5/5×4 = 4:
Answer: x = 4
<span>To solve these GCF and LCM problems, factor the numbers you're working with into primes:
3780 = 2*2*3*3*3*5*7
180 = 2*2*3*3*5
</span><span>We know that the LCM of the two numbers, call them A and B, = 3780 and that A = 180. The greatest common factor of 180 and B = 18 so B has factors 2*3*3 in common with 180 but no other factors in common with 180. So, B has no more 2's and no 5's
</span><span>Now, LCM(180,B) = 3780. So, A or B must have each of the factors of 3780. B needs another factor of 3 and a factor of 7 since LCM(A,B) needs for either A or B to have a factor of 2*2, which A has, and a factor of 3*3*3, which B will have with another factor of 3, and a factor of 7, which B will have.
So, B = 2*3*3*3*7 = 378.</span>