Through how many unit cubes does PQ pass?

[Huszar]

I would be much obliged if someone could help me answer the last 4 questions of an assignment I had to do. If you could type out the solution and answer, I would be grateful. I’ just stuck. It is essential however that you type out the solution (and answer of course) so that I can see/learn what you did.

1. The product of five different odd prime numbers is a five digit number of the form strst, where r = 0 The number of such five digit numbers is:

A) 4 B) 6 C) 8 D) 14 E) 10


2. At one time, the number of employees in a company was a perfect square. Later, with an increase of 100 employees, the number of employees was one more than a perfect square. Now with an additional increase of 100 employees, the total number is again a perfect square. The original number of employees was a multiple of:

A) 5 B) 7 C) 9 D) 11 E) 13


3. In a square array with 10 rows and 10 columns, the element in the m th row and n th column is given by the product (2m-1)(3n-1). The sum of all the elements in the array is between:

A) 7500 and 10 000 B) 10 000 and 12 500 C) 12 500 and 15 000 D) 15 000 and 17 500 E) 17 500 and 20 000

4. Unit cubes (1 x 1 x 1) form a solid rectangular block measuring 20x25x15. Consider the line segment PQ (See attachment visualize.bmp). Through how many unit cubes does PQ pass?

A) 50 B) 58 C) 60 D) 66 E) 70

 

[Muzza]

1) Suppose p_1, \ldots, p_5 are all different, odd primes such that p_1p_2p_3p_4p_5 = strst. Write "strst" = t + 10s + 100 * (0) + 1000t + 10000s, and simplify.

2) Suppose the number of employees is n from the beginning. We have

n = a^2,
n + 100 = b^2 + 1,
n + 200 = c^2

for some natural numbers a, b, c. You want to look at the second and third equations.

3) This can be worked out explicitely (it's an arithmetic series). What is the sum of all elements in row m? Sum that over all rows.

4) Is PQ supposed to be the actual line segment between P and Q, or the red stuff you've drawn on the picture?

 

[Huszar]

1) Suppose p_1, \ldots, p_5 are all different, odd primes such that p_1p_2p_3p_4p_5 = strst. Write "strst" = t + 10s + 100 * (0) + 1000t + 10000s, and simplify.

2) Suppose the number of employees is n from the beginning. We have

n = a^2,
n + 100 = b^2 + 1,
n + 200 = c^2

for some natural numbers a, b, c. You want to look at the second and third equations.

3) This can be worked out explicitely (it's an arithmetic series). What is the sum of all elements in row m? Sum that over all rows.

4) Is PQ supposed to be the actual line segment between P and Q, or the red stuff you've drawn on the picture?

Thanks for the reply.

No, just PQ. I got the picture off google, as I just needed one of a cube. Disregard the red outlining. I didn't quite understand your solution to #1.

Any more replies welcome, and if anyone can work out the answers, again, I'd be grateful.

 

[hando]

hi i have the exact same questions to answer...no joke, what a coincidence...could anyone answer these/go into more detail? I'm terribly short on time and terribly confused...thanks for looking, helping, anytihng

 

[complexPHILOSOPHY]

No one is going to do your homework for you. Go to the homework help section.

 

[hando]

of course not! it's just one of those things where even though you're given a clue, you mull it over for a day but for some reason nothing's clicking. I'm trying to see if someone can help to explain what the previous poster said (granted it's a bit dated), and help me to get this ball rolling

 

[complexPHILOSOPHY]

Again, no one is going to do the homework for you. If you need help or further explanation or require some of the steps to be worked through with you, that is different and perfectly understandable.

Head on over to the homework help section, homie!