Question bank

TMUA-P2-PRF1-001

A direct proof begins: “Suppose $n$ is an even integer. Prove that $n^2$ is even.” Which opening step correctly translat…

TMUA-P2-PRF1-002

Which of the following is a valid direct proof that for odd integers $m$ and $n$, the sum $m + n$ is even?

TMUA-P2-PRF1-003

To prove directly that if integers $a \mid b$ and $b \mid c$ then $a \mid c$, a student writes: $b = as$ and $c = bt$ fo…

TMUA-P2-PRF1-004

A student proves: if $n$ is odd then $n^2 - 1$ is divisible by $8$. Proof: write $n = 2k + 1$; then $n^2 - 1 = 4k(k+1)$;…

TMUA-P2-PRF1-005

Which of the following directly proves that for rational numbers $x$ and $y$, the product $xy$ is rational?

TMUA-P2-PRF1-006

A competition proof claims that for every integer $n$, the number $n^3 - n$ is divisible by $6$, arguing as follows: fac…

TMUA-P2-PRF1-007

A student proves directly that if $n$ is odd then $n^2$ is odd. Which of the following is a valid direct proof?

TMUA-P2-PRF1-008

To prove directly that the product of two odd integers is odd, a student writes $m = 2a+1$ and $n = 2b+1$. Which continu…

TMUA-P2-PRF1-009

Which of the following is a valid direct proof that if $n$ is a multiple of $6$ then $n$ is a multiple of $2$?

TMUA-P2-PRF1-010

Which of the following is a valid direct proof that the sum of two rational numbers is rational?

TMUA-P2-PRF1-011

A student proves directly that $n^2+n$ is even for every integer $n$. Which argument is valid?

TMUA-P2-PRF1-012

Which of the following is a valid direct proof that if $0 < x < y$ then $x^2 < y^2$?

TMUA-P2Y-PRF-007

A direct proof begins: suppose $n$ is odd. Goal: prove $3n + 1$ is even. Which opening correctly translates the hypothes…

TMUA-P2Y-PRF-008

To prove directly that if $n$ is even then $n^2$ is a multiple of $4$, a student writes $n = 2k$ so $n^2 = 4k^2$. Which…

TMUA-P2Y-PRF-009

Which of the following directly proves that if $n$ is odd then $n^2 + n$ is even?

TMUA-P2Y-PRF-010

A competition proof claims: if $3 \nmid n$ then $n^2 = 3t + 1$ for some integer $t$. It argues: either $n = 3k + 1$ or $…