Divisibility, Factors, and Exponents
Stu counted the lines of a page in his book. Counting by threes gave a remainder of two, counting by fives also gave a remainder of two, and counting by sevens gave a remainder of five. What is the least number of lines that could be on that page?
Becky, Ruby, Isabel, Lani, Alma, and Sabrina ran an 800-meter race. Alma beat Isabel by 7 meters. Sabrina beat Becky by 12 meters. Alma finished 5 meters ahead of Lani but 3 meters behind Sabrina. Ruby finished halfway between the first and last woman. In what order did the women finish? What were the distances between them?
Sabrina - 3m - Alma - 3m - Ruby - 2m - Lani - 2m - Isabel - 2m - Becky
A bag contains 25 tickets, each colored either red or yellow. Red tickets are worth $0.50, and yellow tickets are worth $5.00. If the expected value of a ticket drawn at random is $3.20, how many of the tickets are red?
How many of the first one thousand fourth powers end in a one or a six?
Sammy has a sheet of stamps containing 4 rows of stamps, with 8 stamps in each row. Sammy wants to separate all of the stamps by tearing the sheet in as few tears as possible. So, after she makes one tear, she puts the two pieces of the sheet on top of each other before making the next tear. What is the fewest number of tears she can use to get all the stamps separated?
The five-digit number 5ABC9 is the square of an integer where a, b, and c each represent a missing digit. If all five digits are different, what is the five-digit number?
If a and b are integers such that a2 – b2 = 100, what is the greatest possible value of a?
In cents, what is the least total amount that cannot be obtained by using a combination of fewer than eight coins from a collection of pennies, nickels, dimes, and quarters?
If ABCD is a four-digit integer, none of the digits A, B, C, or D are the same, and the equation below is true, what is the value of ABCD? (If AB was the two-digit number 57, then A would equal 5 and B would equal 7) Hint: Neither A nor D is equal to zero.
ABCD x 4 = DCBA
What is the greatest possible integer of k so that 2k is a power of 67!? (! means factorial – 4! is 4*3*2*1 = 24)
Imagine that you have three boxes, one containing two black marbles, one containing two white marbles, and the third, one black marble and one white marble. The boxes were labeled for their contents – BB, WW, and BW – but someone has switched the labels so that every box is now incorrectly labeled. Patrick is allowed to take one marble at a time out of any box, without looking inside, and by this sampling Patrick is able to determine the contents of all three boxes. What is the smallest number of drawings needed to do this? Assume Patrick knows the solution to this problem so he knows what he wants to do.
There are two solutions to the equation: 2x2 + 5x = 12. Name the solutions m and n. What is the value of (m – 1)(n – 1) expressed as a common fraction?