Why isn't the answer to TMUA-P2-PRF1-008 shown upfront?

The options are lettered without marking the correct one so you attempt TMUA-P2-PRF1-008 first — TMUA questions reward spotting the trap before you check the worked solution below.

Do I need a calculator for TMUA-P2-PRF1-008?

No — calculators are banned in the TMUA. TMUA-P2-PRF1-008 is set for a no-calculator target of about 180 seconds; the worked solution shows the fastest by-hand route.

Which TMUA specification point is TMUA-P2-PRF1-008?

TMUA-P2-PRF1-008 is Paper P2 · Direct proof (specification code PRF1), level 2. Drill more of the same topic from this page.

TMUA-P2-PRF1-008 — Paper P2 · Direct proof · Level 2

P2PRF1Level 2~180s target

Spec: PRF1 · No calculator · ~180s

To prove directly that the product of two odd integers is odd, a student writes

m = 2a+1

and

n = 2b+1

. Which continuation validly completes the direct proof?

Options

Check $m = 3$ , $n = 5$ : $mn = 15$ , which is odd, so all such products are odd.
$mn = 4ab+2a+2b = 2(2ab+a+b)$ , which is even, so the product is even.
$mn = 4ab+2a+2b+1 = 2(2ab+a+b)+1$ , which is of the form $2k+1$ and hence odd.
Since odd plus odd is even, the product of two odds is even.

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: direct-proof. Name the trap first, work the stem by hand, then open the solution below.

Show solution

Expanding gives

mn = 2k+1

with

k = 2ab+a+b

an integer, exactly the definition of odd. Trap: the second option drops the final

+1

(an expansion slip that flips the parity), the first offers a single example instead of a general argument, and the last confuses sums with products.

Keep practising

Same topic · same level
P2PRF1Level 2
TMUA-P2-PRF1-003
To prove directly that if integers a | b and b | c then a | c, a student writes: b = as and c = bt for integers s, t, so…
Direct proof
Open TMUA-P2-PRF1-003 →
Same trap: direct-proof
P2PRF1Level 1
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?
Direct proof
Open TMUA-P2-PRF1-007 →
Step up a level
P2PRF1Level 3
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: factorise…
Direct proof
Open TMUA-P2-PRF1-006 →

Browse more

Loading…

Options

Check

m = 3

n = 5

mn = 15

, which is odd, so all such products are odd.

mn = 4ab+2a+2b = 2(2ab+a+b)

, which is even, so the product is even.

mn = 4ab+2a+2b+1 = 2(2ab+a+b)+1

, which is of the form

2k+1

and hence odd.

Since odd plus odd is even, the product of two odds is even.

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

Keep practising

Same topic · same level

P2PRF1Level 2

TMUA-P2-PRF1-003

To prove directly that if integers a | b and b | c then a | c, a student writes: b = as and c = bt for integers s, t, so…

Direct proof

Open TMUA-P2-PRF1-003 →

Same trap: direct-proof

P2PRF1Level 1

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?

Direct proof

Open TMUA-P2-PRF1-007 →

Step up a level

P2PRF1Level 3

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: factorise…

Direct proof

Open TMUA-P2-PRF1-006 →

TMUA-P2-PRF1-008 — Paper P2 · Direct proof · Level 2

Options

Keep practising

Master Direct proof

TMUA-P2-PRF1-008 — Paper P2 · Direct proof · Level 2

Options

Keep practising

Master Direct proof