Bosco

Catherine rolls a standard 6-sided die six times. If the product of her rolls is 60 then how many different sequences of rolls could there have been? (The order of the rolls matters.)    

5 6 2 1 1 1     ₆P₆ = 720    

5 3 4 1 1 1     ₆P₆ = 720    

5 3 2 2 1 1     ₆P₆ = 720    

Add them up.   

By my reckoning, 2160 different sequences.     

_.   

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    

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      

_.    

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.    

_.    

Find a six-digit multiple of $64$ that consists only of the digits $2$ and $4$.   

64 x 3,816 = 244,224     

_.