Careful Counting
We have a group of 3 different dogs and 4 different cats. How many ways are there to arrange these animals in a straight line such that all the dogs are together, and all the cats are together?
288
A679B is a 5-digit number which is divisible by 72. What is the value of A + B?
5
A jar contains 12 marbles: 6 red marbles, 3 white marbles, 2 green marbles and 1 black marble. You are blindfolded. What is the smallest number of marbles you need to remove from the jar to guarantee you end up with at least 3 marbles of the same color?
8
Let N = 2^8 x 3^4 x 7. How many odd factors does N have?
10
Find the value of: 100!/98! × 5!/6!
1650
How many 4-digit integers have all their digits of the same parity; i.e. all even or all odd?
1125
The Greatest Common Factor of the numbers a and b is 10; the Greatest Common Factor of the numbers b and c is 7. What is the smallest possible value of a × b × c?
4900
A 28-mile road is split into three unequal parts. The distance between the centers of the first and third parts is 16 miles. Find the length of the middle portion of the road.
4
How many perfect cube factors does the number 1,000,000,000 have?
16
The Fibonacci sequence is 1, 1, 2, 3, 5, 8, 13, 21, .....; each term after the first two is the sum of the previous two terms. How many of the first 100 terms are odd?
67
How many even integers are there between 200 and 700 whose digits are all different, and the digits come from the list 1, 2, 3, 5, 7, 8?
20
Find GCF (10!, 10^10)
6400
There are 100 soldiers in an army. Every night, 3 of them must stand guard together. Is it possible that after a certain number of nights, every soldier has stood guard with every other soldier exactly once?
This is not possible. If such a night exists, then pick one soldier on that night. He has stood guard with two new soldiers every night - otherwise he wouldn’t have stood guard with each soldier exactly once. But he guards with 2 soldiers every night, so he must have stood guard with an even number of different soldiers. On the other hand, there are 100 total soldiers, so he stood guard with 99 other soldiers, which is a contradiction.
Find the sum of all the factors of 1,200 which have 8 factors of their own.
94
What is the sum of all 2-digit numbers which are exactly 18 more than the sum of their digits?
245