^{399}is divided by 10?

These are all the problems I need help with. I don't get how to do all of them:

1. What is the remainder when (631)^{399} is divided by 10?

2. If July 24 is a Sunday, Them September 24 of the same year will be a _____. (Both July and August have 31 days.)

a)Wednesday b)Thursday c)Friday d)Saturday e)Sunday

3. A printing machine can print 250 flyers per minute. A folding machine can fold the flyers at the rate of 5 per second. If 6 printing machines were being used to print he flyers, then _____ folding machines would be needed to keep up wit the 6 printing machines.

4. Al and Ben set out to buy 300 golf balls, all of the same type. Al bought 120 balls and Ben bought 180. Charlie decides to join them and they decide that all 3 of them will pay an equal amount for the golf balls. Of the $250 Charlie pays, $_____ will go to Al.

5. The chart below show the distances in miles between 4 towns located along a straight road. For instance, the distance between town A and town B is five miles. What is a possible order in which the towns appear along the road?

| A B C D

A | 0 5 5 1

B | 5 0 10 4

C | 5 10 0 6

D | 1 4 6 0

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Parviz F. | Mathematics professor at Community CollegesMathematics professor at Community Colle...

(631 ) ^ 399

Right most digit is going to be 1, and division by 10 will have remainder of 1.

Log ( 631) ^ 399 = 399(log 631)

= 399( 2.8) = ( 1117.2)

1117.2 = 1.12 * 10^3

( 631 ) ^399 = 10 ^ ( 1117.2) / Calculator overflows.

2. Let's make the problem a bit more simple. adding 7 days to the 24th makes it sunday again. Therefore if the number of days between july 24th and sept 24th is evenly divisible by 7 (0 remainder), then the day of the week is Sunday. To calculate the number of days between july 24th and sept 24th take the remaining days in july (7) plus 31 days in august plus 24 days in september. That equals 62. Unfortunately, 62 is not evenly divisible by 7, so it's not sunday. Let's go back to an easy example. Let's add 8 days to july 24th. If we divide 8 by 7, we get 1 remainder of 1. Remainder of 1 makes it monday. Let's add 9 days. If we divide 9 by 7 we get 1 remainder of 2. Remainder 2 makes it tuesday. The pattern we have thus far is remainder 0 = sunday, remainder 1 = monday, remainder 2 = tuesday. When dividing 62 by 7 you get 8 remainder 6 (62 modulus 7 equals 6). Using the pattern you should be able to take it from here.

3. You cannot compare apples to oranges. The rate of operation for the printing machine is stated per minute. The rate of operation for the folding machine is stated per second. The first step is to make the units of measure match. You need to get both rates of operation in either minutes or seconds, it doesn't matter which you pick so long as they are consistent (that being said it is easier to deal in whole numbers rather than fractions). The folding machine folds at a rate of 5 per second. Since there are 60 seconds in a minute, we multiply 5 by 60 to get the rate in minutes. Now we know the folding machine folds 300 per minute. If one printing machine prints 250 per minute that means 6 printing machines will print 6 * 250 = 1500 sheets per minute. The question asks how many folding machine we'll need to keep up, that means we need to fold the same number of sheets per minute as 6 printing machines will print. Therefore:

total sheets printed per minute = total sheets folded per minute

then:

total sheets folded per minute = number of folding machines * number of sheets 1 machine folds per minute

now let's substitute what we know.

1500 = number of folding machines * 300

4. Let's start with what we know. All the golf balls cost the same. The total number of balls bought was 300. Everyone is going to pay the same amount, thus the same number of gold balls. To calculate each person's share we divide the total number of golf balls(300) by the total number of people(3).

300 / 3 = 100.

Charlie's share of the cost is $250 on 100 golf balls Therefore to calculate the cost of each ball is

x = $250/100

x = $2.50

To calculate how much charlie needs to give ben (b) we need to multiply the number of extra balls ben bought (180 minus 100) times the cost per ball ($2.5).

(180 - 100) * $2.5 = b

$200 = b

To calculate how much charlie needs to give al (a) you do the same, just substituting the number of balls al bought.

(120-100) * $2.5 = a

$50 = a

5. This problem is much easier if you draw a line on a piece of paper and plot the points as you go along. Since the towns form a straight line, the easiest thing to do is find the two towns that are furthest from each other. If you look at the chart, that's towns b and c at 10 miles. Draw a line that places them at the end points and make 10 hash marks on the line between the two towns. Then, looking at row B, just plot the points on the line for the remaining towns. Count 5 hash marks from B for town A. Count 4 hash marks from B for town D. If you look at your line you should see:

B - D - A - C

or

C - A - D - B

(depending on how you ordered your end points).

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