Why isn't the answer to TMUA-P2Y-PRF-006 shown upfront?

The options are lettered without marking the correct one so you attempt TMUA-P2Y-PRF-006 first — TMUA questions reward spotting the trap before you check the worked solution below.

Do I need a calculator for TMUA-P2Y-PRF-006?

No — calculators are banned in the TMUA. TMUA-P2Y-PRF-006 is set for a no-calculator target of about 275 seconds; the worked solution shows the fastest by-hand route.

Which TMUA specification point is TMUA-P2Y-PRF-006?

TMUA-P2Y-PRF-006 is Paper P2 · Following proofs (specification code PRF5), level 3. Drill more of the same topic from this page.

TMUA-P2Y-PRF-006 — Paper P2 · Following proofs · Level 3

P2PRF5Level 3~275s target

Spec: PRF5 · No calculator · ~275s

A revision card prints a proof with no title: suppose

m

and

n

are consecutive integers with

m < n

, so

n = m + 1

. Then

m^2 + n^2 = m^2 + (m+1)^2 = 2m^2 + 2m + 1 = 2m(m+1) + 1

. Since one of

m

m + 1

is even, write

m(m+1) = 2t

for some integer

t

; then

m^2 + n^2 = 4t + 1 = 2(2t) + 1

, which is odd. Which title does this proof establish?

Options

The product of two consecutive integers is odd.
The sum of two consecutive integers is even.
Every odd integer is the sum of two consecutive squares.
If $m^2 + n^2$ is odd then $m$ and $n$ are consecutive.
The sum of the squares of two consecutive integers is odd.

Letters only — the answer key is hidden. Work it out, then check the solution below.

Hint (check trope)

No written hint yet — check the trope: proof-direction-reversed, converse-vs-contrapositive. Name the trap first, work the stem by hand, then open the solution below.

Show solution

Read assumption to conclusion: consecutive inputs

\Rightarrow

odd sum of squares. Chain: expand

\Rightarrow

factor

2m(m+1)

\Rightarrow

consecutive parity gives an extra factor

2

\Rightarrow

2(\cdot) + 1

matches the odd definition. Trap: the first two options confuse the sum of squares with the product or sum; the third and fourth state converses and generalizations the proof never attempts.

Keep practising

Same topic · same level
P2PRF5Level 3
TMUA-P2-PRF5-005
A revision guide prints a short proof without a theorem statement: suppose x and y are rational, write x = a/b and y = c…
Following proofs
Open TMUA-P2-PRF5-005 →
Same trap: converse-vs-contrapositive
P2ARG1Level 1
TMUA-P2-ARG1-001
Let P and Q be mathematical statements. The converse of the statement “If P then Q ” is which of the following?
Statements and connectives
Open TMUA-P2-ARG1-001 →
More in this topic
P2PRF5Level 3
TMUA-P2-PRF5-011
Three statements P, Q, R satisfy P ⇒ Q, Q ⇒ R and R ⇒ P. A study group offers five verdicts about the trio. Which is cor…
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Open TMUA-P2-PRF5-011 →

Browse more

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Options

The product of two consecutive integers is odd.

The sum of two consecutive integers is even.

Every odd integer is the sum of two consecutive squares.

m^2 + n^2

is odd then

m

and

n

are consecutive.

The sum of the squares of two consecutive integers is odd.

Letters only — the answer key is hidden. Work it out, then check the solution below.

Keep practising

Same topic · same level

P2PRF5Level 3

TMUA-P2-PRF5-005

A revision guide prints a short proof without a theorem statement: suppose x and y are rational, write x = a/b and y = c…

Following proofs

Open TMUA-P2-PRF5-005 →

Same trap: converse-vs-contrapositive

P2ARG1Level 1

TMUA-P2-ARG1-001

Let P and Q be mathematical statements. The converse of the statement “If P then Q ” is which of the following?

Statements and connectives

Open TMUA-P2-ARG1-001 →

TMUA-P2Y-PRF-006 — Paper P2 · Following proofs · Level 3

Options

Keep practising

Master Following proofs

TMUA-P2Y-PRF-006 — Paper P2 · Following proofs · Level 3

Options

Keep practising

Master Following proofs