MATH 240 EXAM SCRIPT 2026 PRACTICE
SOLUTION FULL QUESTIONS AND CORRECT
ANSWERS GRADED A+
◍ An integer x is odd if.... Ans: there is an integer k such that
x=2k+1
◍ parity. Ans: whether a number is odd or even
◍ r is rational if.... Ans: there exist integers x and y such that y≠0 and
r=x/y where the greatest common divisor of both x and y is 1
◍ an integer x divides integer y iff.... Ans: x≠0 and y=kx for some
integer k
◍ integer n is prime iff.... Ans: n>1 and the only positive integers
that divide n are 1 and n
◍ an integer is composite iff.... Ans: n>1 and there is an integer such
that 1<m<n and m divides n
◍ An integer is either prime or.... Ans: composite
◍ non-negative. Ans: x ≥ 0
, ◍ non-positive. Ans: x <= 0
◍ Allowed assumptions in proofs. Ans: The rules of algebra
- ex: if x, y, and z are real numbers and x = y, then x+z = y+z
The set of integers is closed under addition, multiplication, and
subtraction
- In other words, sums, products, and differences of integers are also
integers
Every integer is either even or odd
If x is an integer, there is no integer between x and x+1
The relative order of any two real numbers
-ex: 1/2<1 or 4.2>3.7
The square of any real number is >= 0
◍ Things in lecture that can be used as assumptions. Ans: - the sum
of two rational numbers x and y is rational
- if n = ab, then a <= sqrt(n) or b <= sqrt(n)
- if a^2 is even, then a is even
SOLUTION FULL QUESTIONS AND CORRECT
ANSWERS GRADED A+
◍ An integer x is odd if.... Ans: there is an integer k such that
x=2k+1
◍ parity. Ans: whether a number is odd or even
◍ r is rational if.... Ans: there exist integers x and y such that y≠0 and
r=x/y where the greatest common divisor of both x and y is 1
◍ an integer x divides integer y iff.... Ans: x≠0 and y=kx for some
integer k
◍ integer n is prime iff.... Ans: n>1 and the only positive integers
that divide n are 1 and n
◍ an integer is composite iff.... Ans: n>1 and there is an integer such
that 1<m<n and m divides n
◍ An integer is either prime or.... Ans: composite
◍ non-negative. Ans: x ≥ 0
, ◍ non-positive. Ans: x <= 0
◍ Allowed assumptions in proofs. Ans: The rules of algebra
- ex: if x, y, and z are real numbers and x = y, then x+z = y+z
The set of integers is closed under addition, multiplication, and
subtraction
- In other words, sums, products, and differences of integers are also
integers
Every integer is either even or odd
If x is an integer, there is no integer between x and x+1
The relative order of any two real numbers
-ex: 1/2<1 or 4.2>3.7
The square of any real number is >= 0
◍ Things in lecture that can be used as assumptions. Ans: - the sum
of two rational numbers x and y is rational
- if n = ab, then a <= sqrt(n) or b <= sqrt(n)
- if a^2 is even, then a is even
