DIVISIBILITY and MODS
What is the remainder of
1*1+2*2+3*3+...+26*26
(mod 5)?
1
Summing up to 25*25 can be grouped into 5 groups which all give the same sum (mod 5), so that's 0 mod 5, and 26*26 = 1 (mod 5).
In how many ways can you line up 5 boys and 5 girls if two kids of the same gender cannot sit next to each other?
(answer = number or expression)
2*5!*5!=28,800
2 = either start with boy or girl
5! ways to order boys and 5! ways to order girls
Evaluate: (4!)4 / (3!)3
(answer = number, not expression!)
1536
Simplifies to 44 * 3! = 3*29=1536
If 3|(a+b) what are all possible remainders of a3+b3 (mod 6)?
0 and 3
Can be seen as divisible by 3 by either observing that a3=a (mod 3), or noting (a+b)3 - (a3+b3) is 3(a2b+ab2). It can clearly be odd or even, so both 0 and 3 are possible.
What is the smallest positive integer with exactly 32 divisors?
(answer = number or expression)
23 * 3 * 5 * 7=840
There are 5 indistinguishable balls and 4 buckets of paints.
In how many ways can you color the balls if exactly one of the colors should not be used?
4*(3+3)=24
4 ways to choose unused color and then it can be 3,1,1 (3 ways) or 2,2,1 (also 3 ways).
Find a formula for the sum of the first N positive numbers which give remainder 2 when divided by 3.
(3N2+N)/2
Rainbow method.
What's the largest number of bishops one can place on a regular chessboard so that no pair attacks each other?
14
There are 15 diagonals in 1 direction; but two of those are 1 square long (opposite corners, which are on a common diagonal in the other direction). Thus 14 is max possible. Top row and bottom row without the corners realizes that maximum (8+6-14).
What are the last two digits of 51*51-26*26?
25
In a soccer league, each team play 5 games; the result can be either a win (3pts), loss (0pts) or a draw (1pt).
How many possibilities are there for the number of points at the end of the season?
15
A team can't reach 11 points, but all other values from 0 to 15 are possible.
There are 21 possible outcomes (in terms of number of wins, draws, and losses) but 3, 4, 5, 6, 7, 9 points can be achieved in 2 ways.
Evaluate (100 choose 50) / (99 choose 49).
2
Evaluate directly. Since (99 choose 49) = (99 choose 50), this also follows from Pascal's identity.
Positive integers using the digits 1, 3, 5, and 7 are written in an increasing order (1, 3, 5, 7, 11, 13, ...). What is the the 90th number written?
1133
There are 4 1-digit numbers, 16 two-digit numbers, 64 3-digit numbers; that's 84 together. The next ones are 1111, 1113, 1115, 1117, 1131, and 1133 is the 90th.
How many divisors of 9,000 are not perfect squares?
40
48 total divisors, of which 8 are perfect squares.
In how many ways can a regular tetrahedron be colored using 3 colors, if two colorings which differ only by a rotation are considered the same. Some colors might be unused.
15
3 ways using 1 color, 3 ways using 3 colors (need to choose the color used for 2 faces), that's 6.
If we are using two colors, it's either 2 and 2 faces each (3 ways -- choose color to skip), or 3 and 1 which is 6 ways (3 ways to chose the color to skip and 2 ways to choose the repeated color).
Evaluate 100*100-99*99+98*98-97*97+...+2*2-1*1.
5050
Simplify 1*1!+2*2!+3*3!+...9*9!
(answer = number or expression)
10!-1