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2023 AMC 12B Problems/Problem 24: Difference between revisions

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Suppose that <math>a,b,c,</math> and <math>d</math> are positive integers satisfying all of the following relations.
Suppose that <math>a,b,c,</math> and <math>d</math> are positive integers satisfying all of the following relations.
What is <math>gcd(a,b,c,d)?</math>
What is <math>gcd(a,b,c,d)?</math>
==Solution==
Denote by <math>\nu_p (x)</math> the number of prime factor <math>p</math> in number <math>x</math>.
We index Equations given in this problem from (1) to (7).
First, we compute <math>\nu_2 (x)</math> for <math>x \in \left\{ a, b, c, d \right\}</math>.
Equation (5) implies <math>\max \left\{ \nu_2 (b), \nu_2 (c) \right\} = 1</math>.
Equation (2) implies <math>\max \left\{ \nu_2 (a), \nu_2 (b) \right\} = 3</math>.
Equation (6) implies <math>\max \left\{ \nu_2 (b), \nu_2 (d) \right\} = 2</math>.
Equation (1) implies <math>\nu_2 (a) + \nu_2 (b) + \nu_2 (c) + \nu_2 (d) = 6</math>.
Therefore, all above jointly imply <math>\nu_2 (a) = 3</math>, <math>\nu_2 (d) = 2</math>, and <math>\left( \nu_2 (b), \nu_2 (c) \right) = \left( 0 , 1 \right)</math> or <math>\left( 1, 0 \right)</math>.
Second, we compute <math>\nu_3 (x)</math> for <math>x \in \left\{ a, b, c, d \right\}</math>.
Equation (2) implies <math>\max \left\{ \nu_3 (a), \nu_3 (b) \right\} = 2</math>.
Equation (3) implies <math>\max \left\{ \nu_3 (a), \nu_3 (c) \right\} = 3</math>.
Equation (4) implies <math>\max \left\{ \nu_3 (a), \nu_3 (d) \right\} = 3</math>.
Equation (1) implies <math>\nu_3 (a) + \nu_3 (b) + \nu_3 (c) + \nu_3 (d) = 9</math>.
Therefore, all above jointly imply <math>\nu_3 (c) = 3</math>, <math>\nu_3 (d) = 3</math>, and <math>\left( \nu_3 (a), \nu_3 (b) \right) = \left( 1 , 2 \right)</math> or <math>\left( 2, 1 \right)</math>.
Third, we compute <math>\nu_5 (x)</math> for <math>x \in \left\{ a, b, c, d \right\}</math>.
Equation (5) implies <math>\max \left\{ \nu_5 (b), \nu_5 (c) \right\} = 2</math>.
Equation (2) implies <math>\max \left\{ \nu_5 (a), \nu_5 (b) \right\} = 3</math>.
Thus, <math>\nu_5 (a) = 3</math>.
From Equations (5)-(7), we have either <math>\nu_5 (b) \leq 1</math> and <math>\nu_5 (c) = \nu_5 (d) = 2</math>, or <math>\nu_5 (b) = 2</math> and <math>\max \left\{ \nu_5 (c), \nu_5 (d) \right\} = 2</math>.
Equation (1) implies <math>\nu_5 (a) + \nu_5 (b) + \nu_5 (c) + \nu_5 (d) = 7</math>.
Thus, for <math>\nu_5 (b)</math>, <math>\nu_5 (c)</math>, <math>\nu_5 (d)</math>, there must be two 2s and one 0.
Therefore,
<cmath>
\begin{align*}
{\rm gcd} (a,b,c,d)
& = \Pi_{p \in \{ 2, 3, 5\}} p^{\min\{ \nu_p (a), \nu_p(b) , \nu_p (c), \nu_p(d) \}} \\
& = 2^0 \cdot 3^1 \cdot 5^0 \\
& = \boxed{\textbf{(C) 3}}.
\end{align*}
</cmath>
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)

Revision as of 17:44, 15 November 2023

Suppose that $a,b,c,$ and $d$ are positive integers satisfying all of the following relations. What is $gcd(a,b,c,d)?$

Solution

Denote by $\nu_p (x)$ the number of prime factor $p$ in number $x$.

We index Equations given in this problem from (1) to (7).


First, we compute $\nu_2 (x)$ for $x \in \left\{ a, b, c, d \right\}$.

Equation (5) implies $\max \left\{ \nu_2 (b), \nu_2 (c) \right\} = 1$. Equation (2) implies $\max \left\{ \nu_2 (a), \nu_2 (b) \right\} = 3$. Equation (6) implies $\max \left\{ \nu_2 (b), \nu_2 (d) \right\} = 2$. Equation (1) implies $\nu_2 (a) + \nu_2 (b) + \nu_2 (c) + \nu_2 (d) = 6$.

Therefore, all above jointly imply $\nu_2 (a) = 3$, $\nu_2 (d) = 2$, and $\left( \nu_2 (b), \nu_2 (c) \right) = \left( 0 , 1 \right)$ or $\left( 1, 0 \right)$.


Second, we compute $\nu_3 (x)$ for $x \in \left\{ a, b, c, d \right\}$.

Equation (2) implies $\max \left\{ \nu_3 (a), \nu_3 (b) \right\} = 2$. Equation (3) implies $\max \left\{ \nu_3 (a), \nu_3 (c) \right\} = 3$. Equation (4) implies $\max \left\{ \nu_3 (a), \nu_3 (d) \right\} = 3$. Equation (1) implies $\nu_3 (a) + \nu_3 (b) + \nu_3 (c) + \nu_3 (d) = 9$.

Therefore, all above jointly imply $\nu_3 (c) = 3$, $\nu_3 (d) = 3$, and $\left( \nu_3 (a), \nu_3 (b) \right) = \left( 1 , 2 \right)$ or $\left( 2, 1 \right)$.

Third, we compute $\nu_5 (x)$ for $x \in \left\{ a, b, c, d \right\}$.


Equation (5) implies $\max \left\{ \nu_5 (b), \nu_5 (c) \right\} = 2$. Equation (2) implies $\max \left\{ \nu_5 (a), \nu_5 (b) \right\} = 3$. Thus, $\nu_5 (a) = 3$.

From Equations (5)-(7), we have either $\nu_5 (b) \leq 1$ and $\nu_5 (c) = \nu_5 (d) = 2$, or $\nu_5 (b) = 2$ and $\max \left\{ \nu_5 (c), \nu_5 (d) \right\} = 2$.

Equation (1) implies $\nu_5 (a) + \nu_5 (b) + \nu_5 (c) + \nu_5 (d) = 7$. Thus, for $\nu_5 (b)$, $\nu_5 (c)$, $\nu_5 (d)$, there must be two 2s and one 0.

Therefore, \begin{align*} {\rm gcd} (a,b,c,d) & = \Pi_{p \in \{ 2, 3, 5\}} p^{\min\{ \nu_p (a), \nu_p(b) , \nu_p (c), \nu_p(d) \}} \\ & = 2^0 \cdot 3^1 \cdot 5^0 \\ & = \boxed{\textbf{(C) 3}}. \end{align*}

~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)

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