Let P be “ n is divisible by 24 ”, Q be “ n is divisible by 8 ”, and R be “ n is divisible by 4 ”. Since 24 = 8 × 3 and 8 = 4 × 2, we have P ⇒ Q and Q ⇒ R. Which verdict is correct?

Worked solution for TMUA-P2-PRF5-007 is revealed on this page behind the solution details.

Why isn't the answer to TMUA-P2-PRF5-007 shown upfront?

The options are lettered without marking the correct one so you attempt TMUA-P2-PRF5-007 first — TMUA questions reward spotting the trap before you check the worked solution below.

Do I need a calculator for TMUA-P2-PRF5-007?

No — calculators are banned in the TMUA. TMUA-P2-PRF5-007 is set for a no-calculator target of about 190 seconds; the worked solution shows the fastest by-hand route.

Which TMUA specification point is TMUA-P2-PRF5-007?

TMUA-P2-PRF5-007 is Paper P2 · Following proofs (specification code PRF5), level 2. Drill more of the same topic from this page.

TMUA-P2-PRF5-007 — Paper P2 · Following proofs · Level 2

P2PRF5Level 2~190s target

Spec: PRF5 · No calculator · ~190s

Let

P

be “

n

is divisible by

24

”,

Q

be “

n

is divisible by

8

”, and

R

be “

n

is divisible by

4

”. Since

24 = 8 \times 3

and

8 = 4 \times 2

, we have

P \Rightarrow Q

and

Q \Rightarrow R

. Which verdict is correct?

Options

$R$ implies $P$ : every multiple of $4$ is a multiple of $24$ .
$Q$ implies $P$ : every multiple of $8$ is a multiple of $24$ .
Nothing links $P$ and $R$ without further hypotheses.
$P$ implies $R$ (by chaining), but $R$ does not imply $P$ (e.g. $n = 4$ ).

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: chain-transitivity, single-counterexample. Name the trap first, work the stem by hand, then open the solution below.

Show solution

Chaining gives

P \Rightarrow R

n = 24k = 4(6k)

. The reverse fails:

n = 4

is divisible by

4

but not by

24

. Trap: running the chain backwards — divisibility chains narrow downwards (larger divisor to smaller), never upwards without extra hypotheses.

Keep practising

Same topic · same level
P2PRF5Level 2
TMUA-P2-PRF5-002
A proof that the square of an odd integer is odd reads: (1) Write n = 2k + 1. (2) Then n^2 = 4k^2 + 4k + 1. (3)??? (4) H…
Following proofs
Open TMUA-P2-PRF5-002 →
Same trap: single-counterexample
P2ARG1Level 3
TMUA-P2-ARG1-009
A textbook exercise states the following facts about positive integers, where P is “ n is divisible by 6 ” and Q is “ n…
Statements and connectives
Open TMUA-P2-ARG1-009 →
Step up a 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 →

Browse more

Loading…

Options

R

implies

P

: every multiple of

4

is a multiple of

24

Q

implies

P

: every multiple of

8

is a multiple of

24

Nothing links

P

and

R

without further hypotheses.

P

implies

R

(by chaining), but

R

does not imply

P

(e.g.

n = 4

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

Keep practising

Same topic · same level

P2PRF5Level 2

TMUA-P2-PRF5-002

A proof that the square of an odd integer is odd reads: (1) Write n = 2k + 1. (2) Then n^2 = 4k^2 + 4k + 1. (3)??? (4) H…

Following proofs

Open TMUA-P2-PRF5-002 →

Same trap: single-counterexample

P2ARG1Level 3

TMUA-P2-ARG1-009

A textbook exercise states the following facts about positive integers, where P is “ n is divisible by 6 ” and Q is “ n…

Statements and connectives

Open TMUA-P2-ARG1-009 →

Step up a 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 →

TMUA-P2-PRF5-007 — Paper P2 · Following proofs · Level 2

Options

Keep practising

Master Following proofs

TMUA-P2-PRF5-007 — Paper P2 · Following proofs · Level 2

Options

Keep practising

Master Following proofs