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MATH / Lizzy, Megan, Oscar, and Patrick each have x pieces of candy, where x is a positive integer. Unfortunately, Patrick is the only one of the four who likes candy. So Lizzy gives all …

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problem

Lizzy, Megan, Oscar, and Patrick each have xx pieces of candy, where xx is a positive integer. Unfortunately, Patrick is the only one of the four who likes candy. So Lizzy gives all her candy to Megan. Then Megan gives all the candy she now has (which includes the candy Lizzy gave her) to Oscar. Then Oscar gives all the candy he now has to Patrick. Let PP be the number of pieces of candy Patrick has in the end. How many of the following statements are true? (Assume that we do not know exactly what xx is.) (a) 22 can be a divisor of PP. (b) 22 must be a divisor of PP. (c) 33 can be a divisor of PP. (d) 33 must be a divisor of PP. (e) 44 can be a divisor of PP. (f) 44 must be a divisor of PP.
Plain-text mathematical notation (without MathML)
Lizzy, Megan, Oscar, and Patrick each have x pieces of candy, where x is a positive integer. Unfortunately, Patrick is the only one of the four who likes candy. So Lizzy gives all her candy to Megan. Then Megan gives all the candy she now has (which includes the candy Lizzy gave her) to Oscar. Then Oscar gives all the candy he now has to Patrick.

Let P be the number of pieces of candy Patrick has in the end. How many of the following statements are true? (Assume that we do not know exactly what x is.)

(a) 2 can be a divisor of P.
(b) 2 must be a divisor of P.
(c) 3 can be a divisor of P.
(d) 3 must be a divisor of P.
(e) 4 can be a divisor of P.
(f) 4 must be a divisor of P.
Original LaTeX notation
Lizzy, Megan, Oscar, and Patrick each have $x$ pieces of candy, where $x$ is a positive integer. Unfortunately, Patrick is the only one of the four who likes candy. So Lizzy gives all her candy to Megan. Then Megan gives all the candy she now has (which includes the candy Lizzy gave her) to Oscar. Then Oscar gives all the candy he now has to Patrick.

Let $P$ be the number of pieces of candy Patrick has in the end. How many of the following statements are true? (Assume that we do not know exactly what $x$ is.)

(a) $2$ can be a divisor of $P$.
(b) $2$ must be a divisor of $P$.
(c) $3$ can be a divisor of $P$.
(d) $3$ must be a divisor of $P$.
(e) $4$ can be a divisor of $P$.
(f) $4$ must be a divisor of $P$.

level

Level 5

type

Number Theory

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Source and history

task_id
5c92ecb2-104b-5354-bd1f-8f6561951eaf
task_key
number~5ftheory--test--5c92ecb2-104b-5354-bd1f-8f6561951eaf
task_revision_id
1
config / split
number_theory / test

Official source

initial import