Marvelous Mods & Devious Digits
What is the remainder when 4386 is divided by 5?
4
How many distinct ways are there to arrange 5 identical blue blocks and 3 identical orange blocks in a straight line?
56
a679b is a five-digit number divisible by 72. What is the value of a+b?
5
The sequence 1, 1, 2, 3, 5, 8, . . . is called the Fibonacci sequence. Each successive term is the sum of the previous two terms. How many odd numbers are there among the first 1,000 terms of the sequence?
667
Find the largest power of 10 that divides 100! (that is, 100 factorial).
1024
Emily must select four courses to take from the list: English, Algebra, Geometry, History, Art, and Latin. She must take English and at least one mathematics course. In how many ways can she select four classes to take?
9
What is the sum of the three numbers less than 1000 that have exactly five positive integer divisors?
722
How many ordered triples (p, q, r) of three prime numbers exist for which the sum p + q + r is 24? (order matters!)
15
How many two digit numbers are there in which the tens digit is divisible by the unit digit?
23
In a weekly class of 18 students, a teacher always selects two students to clean up the room after class. If each student can be selected at most twice, for how many weeks can the teacher select students for cleanup without repeating the same pair?
18
What is the greatest divisor of 1800 that is not a multiple of 30?
225
One bell rings every 4 hours, a second bell rings every 6 hours, and a third bell rings every 10 hours. If all three bells just rang together, how many times in the next 100 hours will exactly two bells ring together (including the 100th hour)?
13
Find the last digit of the product of all odd numbers from 101 through 999.
5
A store sells red, white, blue, and yellow beads. In how many ways can I purchase a group of 6 beads, of which at least two are red and no more than one is yellow?
25
Vini wants to write a 6-digit number that consists of distinct digits, is divisible by 4, 5, and 9, and doesn’t use the digits 7 or 9. In how many ways can he do this?
144
Alex and Ben were backpacking when some hungry traveler asked them for some food. Alex had 3 lbs of potatoes and Ben had 2 lbs. They decided to take a break, make potato soup and split it equally three ways with the traveler. The grateful traveler paid $4 for the food. If Alex and Ben divide the money fairly (based on their potato contribution), how much money does Alex get?
$3.20