Bosco

My answer is that it cannot be done.   

First, I intuited this because both 52 and 39 are multiples of 13.   

Any subtaction of (52 • n₁) – (39 • n₂) will leave a multiple of 13.   

Second, I graphed 52x – 39y = 1 on Desmos.  I expanded the gridlines so that the  

distance between each large division was 1.  This was just to make it easier to read.  

I checked the path of the curve to see if it passes through any intersection of x and y  

perfectly, up to x = 20, because any x > 19 exceeds the 1,000 candies in the supply.  

The curve does not pass perfectly through any intersection.  This means that there  

is no solution that consists of two integers. 

_.   

At a cafeteria, Mary orders two pieces of toast and a bagel, which comes out to $3.15. Gary orders a bagel and a muffin, which comes out to $3.50. Larry orders a piece of toast, two bagels, and three muffins, which comes out to $8.15. How many cents does one bagel cost?    

                                        T + B = 315¢    ==>>    T = 315 – B    

                                        B + M = 350¢    ==>>    M = 350 – B    

                                         T + 2B + 3M = 815¢   

                                         (315 – B) + 2B + [3 • (350 – B)]  =  815    

                                         315 – B + 2B + 1050 – 3B  =  815    

                                          – 2B + 1365  =  815    

                                          – 2B  =  – 550    

                                               B  =  275¢   

_.   

The total couldn't be $1482.75 because when you divide by 99, you don't get an even amount for each book.  

1482.75 / 99 = 14.9772727272727272727•••    It's unlikely that you'd find that on the Barnes & Noble price tag.  

It's also an unusual assumption in "There are two possible pairs of digits that have a sum of 6: 0 and 6, and 1 and 5."   

6+0 and 1+5 do indeed toal 6.  But so do 2+4, 3+3, 4+2, and 5+1.  Check my answer ... 4 and 2 are the correct pair.

_.   

A school orders  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?  a482.7b     

The other problem like this says that the school orders 99 textbooks.     

This problem has a blank spot where that 99 is in the other problem.     

So, I will assume the number of books in this problem should be 99.    

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.    

4 + 8 + 2 + 7 = 21 so the sum of the two missing     

numbers will have to add 6 (to make 27, which is     

evenly divisible by 9) or 15 (to make 36) to the 21.     

I found that 4482.72 is evenly divided by 99.   

So the missing numbers are a = 4 and b = 2.    

And the cost of each textbook was 45.28.    

_.   

I made a conversion mentally that I now feel I should have shown explicitly.  

The problem asks for 1/x + 1/y = 1/123         with x < y    

I submitted 1/369 + 2/369 = 3/369 = 1/123  

For clarity, there should have been a step before that.    

                  1/369 + 1/184.5    

Stated thus, the x and y positions are reversed.    

Pedantically, I admit I should not have done that.   

I just thought the equation looked nicer as 1, 2, 3. 

x = 184.5 and y = 369     note that x < y   

check answer       

1/x        + 1/y       =  1/123    

1/184.5 + 1/369  =?  1/123      

2/369    + 1/369  =  3/369  =  1/123       True    

_.   

What is the only solution to 1/x + 1/y = 1/123 where x < y?    

You could go for 3 / (3)(123) which would be    

written as 1 / 369 + 2 / 369 = 3 / 369 = 1 / 123     

but I don't guarantee that it's the only solution.   

_.   

You have a total supply of $1000$ pieces of candy, and an empty vat.  You also have a machine that can add exactly $7$ pieces of candy per scoop to the vat, and another machine that can remove exactly $6$ pieces of candy with a different scoop from the vat.  When these two machines are done, there is only one piece of candy left in the vat.  What is the smallest possible number of times the the first machine added candy to the vat?    

     One      

     Let the 1st machine cycle one time .....  7 pieces added    

     Let the 2nd machind cycle one time ....  6 pieces removed     ==>> 1 piece remaining in vat    

     _.         

A positive integer is called terrific if it has exactly 5 positive divisors. What is the smallest number of primes that could divide a terrific positive integer?  

The smallest number of prime divisors is one.     

As an example, 16 has 5 positive divisors:  1, 2, 4, 8, 16.  2 is a prime number.    

_.   

A terminal zero is a $0$ that appears at the end of a number. For example, the number $3,800$ has two terminal zeros.    
How many terminal zeroes does $40 \cdot 6 \cdot 75 \cdot 12$ have?    

     40 • 6 • 75 • 12  =  216,000 ... so. three terminal zeroes.      

_.   

The number $100$ has four perfect square divisors, namely $1,$ $4,$ $25,$ and $100.$     
What is the smallest positive integer that has exactly $2$ perfect square divisors?    

That would be four ... it has two perfect square divisors, namely one and four.    

_.