Students in the class of Peter practice the addition and multiplication of integer numbers.The teacher writes the numbers from $1$ to $9$ on nine cards, one for each number, and places them in an ballot box. Pedro draws three cards, and must calculate the sum and the product of the three corresponding numbers. Ana and Julián do the same, emptying the ballot box. Pedro informs the teacher that he has picked three consecutive numbers whose product is $5$ times the sum. Ana informs that she has no prime number, but two consecutive and that the product of these three numbers is $4$ times the sum of them. What numbers did Julian remove?
Problem
Source: Cono Sur 2002 P1
Tags: number theory, consecutive, Sums and Products, cono sur
MathClassStudent
01.08.2018 04:08
wow cool problem
NikoIsLife
01.08.2018 04:20
$(3,4,6)$ or $(2,6,7)$
Pedro must have gotten $5$ in order for the product to be divisble by $5$. Let the other two number be $a,b$. We have:
$$5ab=5(a+b+5)\implies(a-1)(b-1)=6$$We have these possible solutions:
$$(a,b)=(2,7),(3,4)$$
Ana informed that she has no prime number, but two consecutive. The only way this could happen is if she had $(8,9)$. Let her third number be $c$.
$$72c=4(c+17)\implies c=1$$
Hence, there are two possible answers: $(3,4,6)$ or $(2,6,7)$.
SeanTran
30.06.2020 08:49
Quote: Pedro must have gotten $5$ in order for the product to be divisble by $5$. Let the other two number be $a,b$. We have: $$5ab=5(a+b+5)\implies(a-1)(b-1)=6$$We have these possible solutions: $$(a,b)=(2,7),(3,4)$$ Pedro has 3 consecutive numbers. Those 3 must be 3, 4, and 5. Ana then takes 1, 8, and 9. Thus, there is only one answer of (2, 6, 7).