Given integers: Team R scored -215, Team S scored a negative multiple of 5 that is greater than -215, Team T scored the smallest positive composite less than 20. List all possible S scores and T score, then rank teams best→worst.
S: negative multiples of 5 greater than -215 are -210,-205,…,-5. T: smallest positive composite <20 is 4. Ranking best→worst: Team R (most negative), any S (less negative than R), Team T (positive, worst).
Point A = -41. Point B is a positive integer whose square is <1600 and whose distance to zero is less than |A|. Point C is an integer midpoint of A and B. List all B and corresponding integer C values.
B: positive integers with square <1600 → B≤39, distance<41 → B∈{1..39}. C=(A+B)/2 must be integer so A+B even → B odd (since A=-41 odd). So B∈{1,3,5,…,39}; C values = (-41+B)/2 corresponding to each B.
Find all negative integers N such that -80 < N < -20 and N is divisible by both 8 and 10.
LCM(8,10)=40. Multiples of 40 in range: -40 only. So N = -40.
A real x satisfies 2 < x < 3, x is algebraic of degree 2 (root of a quadratic with integer coefficients), and its decimal expansion does not terminate nor repeat. Is x rational or irrational? Explain.
Irrational. Algebraic degree 2 non-integer roots (quadratic irrationals) are irrational; their decimal expansions are nonterminating, nonrepeating.
Point P is between 4 and 5 and equals p/q in lowest terms with q≤20. List three possible p/q values and explain why P is rational.
Examples: 9/2 (4.5), 19/4 (4.75), 21/5 (4.2). Each is a ratio of integers ⇒ rational.
Three leaderboard scores: A=-512, B is negative and a power of 2 greater than A, C is positive and equals the sum of digits of |A|. Give possible B and C and order best→worst.
Powers of 2 greater than -512 (less negative): -256,-128,-64,…; B could be those negatives. Sum of digits of |A|=5+1+2=8 so C=8. Order best→worst: A (-512), any B (e.g., -256), C (8).
Compare -e and -2.718 using a < or > and explain (use known decimal for e ≈ 2.7182818…).
-e < -2.718 because -2.7182818… < -2.718, so -e is slightly smaller.
Find all integers k such that -15 < k ≤ 15 and |k| is a prime. List them and explain.
Primes: 2,3,5,7,11,13 → k can be ±2,±3,±5,±7,±11,±13 (within range) so set = {-13,-11,-7,-5,-3,-2,2,3,5,7,11,13}.
A decimal 0.212121… is it rational? Express as a fraction in lowest terms.
Yes rational. 0.212121… = 21/99 = 7/33.
Point X is irrational between -1 and 0. Give two standard examples and justify.
Examples: -√2/2 ≈ -0.7071, -π/4 ≈ -0.7854. Both are nonterminating, nonrepeating decimals and cannot be written as ratios of integers.
Find all integer midpoints m between -100 and 100 such that m is integer midpoint of two integers a and b where a=-73 and b> a and b<0.
Midpoint m=(a+b)/2 integer ⇒ a+b even ⇒ -73 + b even ⇒ b odd. b ranges from -72 to -1 odd → b∈{-71,-69,…,-1}. m values = (-73+b)/2 produce integers from (-73+ -71)/2=-72 to (-73+ -1)/2=-37. So m∈{-72,-70,…,-37}.
Is the decimal 0.12345678910111213… (concatenation of positive integers) rational or irrational? Explain.
Irrational. It’s the Champernowne-like constant: nonterminating and nonrepeating, so not expressible as a ratio of integers.
Write three unequal rational numbers between -2.001 and -1.999.
Examples: -2.0009, -2.0001, -1.9995 (all rational decimals).
Let a number r equal √(m/n) where m and n are integers and m/n is not a perfect square. If m/n ∈(1,4), is r rational or irrational?
Irrational. Square root of non-perfect-square rational is irrational (quadratic irrational).
Compare 355/113 and π using < or > and justify (use known approximation 355/113≈3.14159292).
355/113 > π slightly because 355/113≈3.14159292 while π≈3.14159265; thus 355/113 − π >0 (very small).
List all integers z such that -9 < z < 9 and gcd(z,12)=3.
Integers with gcd 3 with 12 are numbers divisible by 3 but not by 4. Multiples of 3 in range: -8..8 → -6,-3,0,3,6. Exclude those divisible by 4? 0 divisible by 4 and 12 so gcd(0,12)=12 not 3, so remove 0. Check -6 gcd(6,12)=6, not 3; -3 gcd=3 ok; 3 gcd=3 ok; 6 gcd=6 no. So final: -3,3.
A number x satisfies: x is rational, 0<x<1, and decimal expansion has period 8 (repeating every 8 digits). Give general form and one example.
Any such x = p/(10^8−1)·k for integers p,k simplifying. Example: 0.12345678 repeating = 12345678/99999999 reduces if possible.
Given integers a=-21 and b=45, compute midpoint and state whether midpoint is rational and why.
Midpoint = (a+b)/2 = (24)/2 =12, rational (integer).
Find all multiples of 11 between -100 and 100 that are palindromic as 2-digit or 3-digit positive numbers when absolute value taken.
Positive palindromic numbers that are multiples of 11: 11,22,33,44,55,66,77,88,99. Negatives corresponding: -11,-22,…,-99 within range. So list those (±11..±99).
A decimal 0.(142857) repeating equals what fraction? Prove it’s rational.
0.142857 repeating = 142857/999999 = 1/7.
Let s be an irrational in (0,1). Is 1−s rational or irrational? Explain with example.
1−s is irrational. Example: s=√2−1 (irrational), 1−s=2−√2 (still irrational). Difference of rational and irrational is irrational.
Find integer solutions n where n^2 < 200 and n is negative: list n values.
n negative integers with square <200: n ∈ {−1,−2,…,−13} since 13^2=169,14^2=196? Wait 14^2=196 <200 so include -14; 15^2=225>200. So n = -1 to -14.
Compare 0.999… and 1 using <, >, or = and justify.
0.999… = 1 (they are equal).
Find three distinct rational numbers whose decimal expansions terminate and lie strictly between √3 and 2.
√3≈1.732…, examples: 1.74, 1.8, 1.9 (all rational terminating decimals).
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