Bosco

The product of a set of distinct positive integers is $630$.  

If one of the numbers is 42, what are the other two numbers?  

630 divided by 42 is 15.  The factors of 15 are 3 and 5.    

_.   

One ordered pair $(a,b)$ satisfies the two equations $ab^4 = 48$ and $ab^3 = 24$. What is the value of b in this ordered pair?   

                                                     ab⁴  =  48       

                                                     ab³  =  24    

ab⁴ = 48 and ab³ = 24 so let's   

divide both sides of the first by     

respective equal values of the     

second to obtain                              b = 2     

the problem doesn't ask for     

the value of a, but it's                       a = 3      

_.    

                                     x^2 - 5x + 4 = -x^2 - 25x - 14    

combine like terms        2x² + 20x +18  =  0    

divide by 2                      x² + 10x + 9  =  0    

factor                              (x + 1)(x + 9)  =  0    

                                                 (x + 1)  =  0      ==>>  x = –1    

                                                 (x + 9)  =  0      ==>>  x = –9    

                                                         x  =  –1, –9   

_.   

A school orders $99$ textbooks, all for the same price.   

When the bill for the total order comes, the first and last   

digits are obscured. What are the missing digits?   
$_5801.3_   

Consider that the price per book is multiplied by 99.   

And, multiplying by 99 incorporates multiplying by 9.    

The digits of the product of a number multiplied by 9    

will always sum to a total that is evenly divisible by 9.    

5 + 8 + 0 + 1 + 3 = 17  so the sum of the two missing     

numbers will have to add 1 or 10 or 19, etc. to that 17.     

Using brute force, I reckoned that 15801.39 is evenly

divided by 99.  So the missing numbers are 1 and 9.    

And the cost of each textbook was 159.61.    

_.    

In order not to have two yellows next to each other,   

the three yellows must occupy squares 1, 3, and 5.  

That leaves squares 2 and 4 to fill, and two colors   

to do it with. There are only four ways.  

Blue Blue     Red Red     Blue Red     Red Blue   

So, the answer is four ways.    

_.   

What you did was take the long way around to    

say that two times zero equals one times zero.    

And then you divided by zero to obtain 1 = 2.     

_.   

                                                    x^2 - 5x + 4 = -x^2 - 25x - 14   

                                                    2x² + 20x + 18  =  0     

                                                      x² + 10x + 9  =  0    

                                                      (x + 1)(x + 9)  =  0    

                                                                 (x + 1) = 0     ==>>     x = –1    

                                                                 (x + 9) = 0     ==>>     x = –9    

                                                                  x = –1, –9    

_.