A
What is the modulo 13 residue of 247+5·39+7·143+4·15?
8
Express 0.5 as a common fraction in lowest terms.
Ans: 5/9
Expand the product 4(x − 5)(x + 8).
Ans: 4x^2 + 12x − 160
Given that k is a positive integer less than 6, how many values can k take on such that 3x ≡ k (mod 6) has no solutions in x?
Ans: 4
A palindrome is a number that is the same when read forwards and backwards, such as 43234. How many positive 3-digit palindromes are multiples of 3?
30
How many positive and negative integers is 12 a multiple of?
12. there are 6 positive divisors and 6 negative divisors
What is the units digit of the sum of the squares of the first 2007 odd, positive integers?
5
6. What is the sum of the greatest common divisor of 30 and 81 and the least common multiple of 36 and 12? Ans: 39
39
7. Karlanna places 600 marbles into m total boxes such that each box contains an equal number of marbles. There is more than one box, and each box contains more than one marble. For how many values of m can this be done?
22
9. Find the positive base b in which the equation 4 · 12 = 103 is valid.
5
16. What is the maximum value of 4(x + 7)(2 − x), over all real numbers x? Ans: Find average of -7 and 2, which is 2.5. 4*4.5*4.5 = 81
81
14. What is the smallest positive integer n such that 3n is a perfect square and 2n is a perfect cube?
Ans: n = 3^3 · 2^2 = 108
11. Suppose (b_a)^2 = 71_a, where a and b represent two distinct digits. If b = a − 1, find a.
9
If ⌈x⌉+⌊x⌋+x = 4.8, solve for x.
1.8
19 see picture
91
There are 20n members in the Trumpington marching band, and when they line up in rows of 26, there are 4 band members left over. If n is an integer and there are fewer than 1000 band members, what is the maximum number of people that could be in the Trumpington marching band?
940
17. If 3x+y = 81 and 81x−y = 3, then what is the value of the product xy? Express your answer as a common fraction
255/64
20 .The speed of a stream is 3 km per hour. A boat travels upstream 12 km and then returns to its original position downstream along the same route. If the speed of the boat in still water is 9 km per hour, what is the average speed of the boat for the entire round trip?
8
Compute (29^13)−(5^13) mod 7
3
4. Compute 997−1 modulo 1000. Express your answer as an integer from 0 to 999. Ans: 997 ≡ −3 mod 1000 =⇒ (−3)·333 = −999 ≡ 1 mod 1000 =⇒ 333
Ans: 333