Can You Mod This?
If 485 is equivalent to 5 (mod n) and n > 1, how many positive ODD numbers are possible values of n?
We want odd positive divisors of 480 that are greater than 1. There are 3 of them.
If 5n is a multiple of 6, and 6n is a multiple of 7, what is the least possible value of 7n ?
6 * 7 * 7 = 294.
How many pairs of perfect squares differ by 80?
Factor the difference of squares to find 3 pairs:
212, 192 ,
122, 82,
92, 12.
In how many distinct ways can four identical red chips and two identical white chips be arranged in a circle? Two results that are rotations of each other are considered equivalent.
There can be 0,1,2,3, or 4 red chips between the two white ones, but because rotations are equivalent, 0 is equivalent to 4, and 1 is equivalent to 3. Hence the answer is 3.
A drawer contains 8 grey socks, 5 white socks, and 10 black socks. If socks are randomly taken from the drawer without replacement, how many must be taken to be sure that 4 socks of the same color have been taken?
By Pigeonhole, 10.
A broken chatbot begins reciting the alphabet over and over again, starting with the letter A. What is the 326-th letter that the chatbot recites?
Powers of 3 (mod 26):
3, 9, 1, 3, 9, 1...
Therefore 3^26 mod 26 = 9. The 9th letter of the alphabet is I.
Positive integers a and b satisfy the following conditions:
(1) a>b. (2) gcd(a,b) = 8. (3) lcm(a,b) = 96.
What is the least possible ratio of a to b?
Both a and b contain 23. To obtain 96 = 25*3, we need to assign the prime numbers 2, 2, and 3 to a and b, in some combination.
4:3 or 4/3
Simplify the expression below.
2/(1-1/(1-sqrt2))
Use multiplication by conjugates to obtain
2 - sqrt2
How many 3-letter permutations can be formed from the letters in ALASKA?
1 distinct letter (AAA): 1 solution.
2 distinct letters: 3*3 = 9.
3 distinct letters: permutations of 3 letters chosen from 4: 6*4 = 24.
In total, 34.
Find the value of digit A if the five-digit number 12A3B is divisible by 36, and A does not equal B.
B is 2 or 6. But if B = 6, then A = 6. Hence B = 2.
Then A = 1.
What are the last three digits of 9989 ?
(-2)9=-512, so we have
1000-512=488
Let N = 3374135. How many divisors N meet at least one of the following conditions?
(1) The divisor is a multiple of 7.
(2) The divisor ends in a 7.
N has 4*4*6 = 96 divisors that are multiples of 7.
The divisors that are not multiples of 7, but end in 7, can be found by looking at powers of 13==3 (mod 10). We find 6 instances that work.
96 + 6 = 102.
A number is five greater than its reciprocal.
How much greater than the cube of its reciprocal is the cube of the number?
Expand LHS of
(x-1/x)^3=5^3 .
We find that
125=x^3-1/x^3-3(1-1/x)
The answer is 140.
Four traveling friends decide to spend the night in a motel. The motel has four identical (indistinguishable) rooms that the friends might occupy. If each room sleeps at most three people, how many different ways are there to place the friends in the rooms?
Do cases. For {3,1} (i.e. three people in one room, one person in another), 4C3 = 4 solutions.
For {2,2}, 4C2/2 = 3 solutions. For {2,1,1}, 4C2 = 6 solutions. For {1,1,1,1}, 1 solution. In all, that's 14.
Using two straight lines, divide the face of a standard 12-hour clock into three parts so that the sum of numbers in each of three parts is the same as in any other part.
3 ‘bands’ each sum to 26: {11, 12, 1, 2}, {9,10,3,4}, {5,6,7,8}