DIVISIBILITY
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.
How many diagonals does a regular n-gon with interior angles of 162 degrees have?
170
162*n=(n-2)*180
n=20
So, we have a 20-gon. How many diagonals?
From each of the 20 vertices, exit 17 diagonals. We counted each twice, so (20*17)/2=170
What is the remainder when 456,564,465,645 is divided by 6?
(answer = number or expression)
3
It is divisible by 3 (from the sum of digits) but not even, so its remainder (mod 6) has to be 3.
In a 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?
We can go for 18 weeks via (1,2), (2,3), …, (17, 18), (18,1). We can’t go for more weeks because then we’d use more than 36 students and by pigeonhole, at least one student would be used at least 3 times.
How many perfect cube divisors does 1,000,000,000 have?
16
109 = 29*59. For perfect cubes, we need combinations of (20, 23, 26, 29) with (50, 53, 56, 59).
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
The sequence goes O, O, E, O, O, E, so there are 666 odd numbers among the first 999 and the 1000th is also odd.
What is the measure of the interior angle in a regular pentagram?
36
The little triangle is isosceles and the two equal angles are complementary to the interior angle of a regular pentagon which is 31805=108. Thus these angles are 180-108=72and the angle we seek is 180-272=36.
A679B is a five-digit number divisible by 72. What is the value of A+B?
5
We need divisibility by 8 and 9, so the last 3 digits must be divisible by 8, which gives B=2. The sum of digits must be divisible by 9, which gives us the value of A = 3.
In how many ways can a chess knight move from the lower left corner of a standard (8x8) chess board to the upper right corner, if it can only move up and to the right?
0
Each knight move adds either 2+1 or 1+2 to the sum of horizontal and vertical positions. We start at (1,1) and end at (8,8). We cannot move from sum=2 to sum=16 by steps of 3.
What is the greatest divisor of 1800 that is not a multiple of 30?
225
1800 = 23*32*52. We must exclude divisors that have a 2, a 3, and a 5 all present, so the choice is between “biggest without 2”, “biggest without 3” and “biggest without 5”.
What is the sum of all positive divisors of 72?
195
72=2332, so the divisors are 1,2, 4, 8 and 3 those and 3 those. Hence the sum is(1+2+4+8)(1+3+9)=15*13=195
What is the largest number of interior right angles that a 7-gon can have?
5
If it has 6, the sum of angles would be more than 690+360=900 but it must be 5180=900
How many ordered triples of three prime numbers exist for which the sum of the members of the triple is 24?
15
One of the primes has to be 2 (otherwise the sum would be odd). (2,3,19)x6, (2,5,17)x6; (2,11,11) x3
In Harry Potter’s world, wizarding money comes in three coins: bronze Knuts, silver Sickles, and golden Galleons. 29 Knuts make up one Sickle, and there are 17 Sickles in a Galleon. In how many ways can 2 Galleons be made as a combination of Galleons, Sickles and Knuts?
54.
Let’s split into cases, with different numbers of Galleons in each case. The Galleons will make for some part of the sum; for the rest, let’s see what is the maximal number of Sickles that fit, and then we’ll replace these Sickles one by one by Knuts. Case 1: just 2 Galleons, no other coins, 1 way. Case 2: 1 Galleon, 18 ways (G+17S, G+16S+knuts, G+15S+knuts etc.). Case 3: 0 Galleons, 35 ways (34S, 33S+knuts, 32S+knuts, etc.). The total is 1+18+35=54.
What is the sum of the three numbers less than 1000 that have exactly five positive integer divisors?
722
In how many ways can you cover a 25 shape with dominoes? [You are not allowed to rotate, so ==| and |== are considered distinct.]
8
We can have two horizontal dominoes:
0 times → 1 solution
1 time → 4 solution (start a position 1, 2, 3, or 4)
2 times → 3 solutions (the single | is either leftmost, in between or right most)
Total is 1+4+3=8.
What is the area of a regular 12-gon inscribed in a circle with radius 10?
300
Consider the quadrilateral ABCO where O is the center of the circle. Then |OA|=|OC|=AC=10, since OAC is one sixth of the regular hexagon centered at O; hence an equilateral triangle. However, |OB|=10, since it’s also the radius and OB is equal to the sum of the lengths of altitudes to AC in OAC and BAC, so the area of the quadrilateral OABC is |OB||AC|2=50 and so the total area is 650=300.