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代数因式分解 + 质数整数因数分析 Prime Integer Divisor[c in progress]

作者：

radmin

在

Q1

Integers $a$, $b$, and $c$ satisfy $ab + c = 100$, $bc + a = 87$, and $ca + b = 60$. What is $ab + bc + ca$?

$\textbf{(A) }212 \qquad \textbf{(B) }247 \qquad \textbf{(C) }258 \qquad \textbf{(D) }276 \qquad \textbf{(E) }284 \qquad$

Q2

The number of triples $(a, b, c)$ of positive integers which satisfy the simultaneous equations

$ab+bc=44$
$ac+bc=23$

is

$\mathrm{(A) \ }0 \qquad \mathrm{(B) \ }1 \qquad \mathrm{(C) \ } 2 \qquad \mathrm{(D) \ }3 \qquad \mathrm{(E) \ } 4$

Q3 1997 AHSME Problems/Problem 28

How many ordered triples of integers $(a,b,c)$ satisfy $|a+b|+c = 19$ and $ab+|c| = 97$?

$\textbf{(A)}\ 0\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 12$

Q4

Three positive integers are each greater than $1$, have a product of $27000$, and are pairwise relatively prime. What is their sum?

$\textbf{(A)}\ 100\qquad\textbf{(B)}\ 137\qquad\textbf{(C)}\ 156\qquad\textbf{(D)}\ 160\qquad\textbf{(E)}\ 165$

Q5

How many ordered triples $(x,y,z)$ of positive integers satisfy $\text{lcm}(x,y) = 72, \text{lcm}(x,z) = 600,$ and $\text{lcm}(y,z)=900$?

$\textbf{(A)}\ 15\qquad\textbf{(B)}\ 16\qquad\textbf{(C)}\ 24\qquad\textbf{(D)}\ 27\qquad\textbf{(E)}\ 64$

Q6 AIME show your work

Find the number of ordered triples $(a,b,c)$ where $a$, $b$, and $c$ are positive integers, $a$ is a factor of $b$, $a$ is a factor of $c$, and $a+b+c=100$.

Q7 AIME show your work 2021 AIME II Problems/Problem 7

Let 

$a, b, c,$

 and 

$d$

 be real numbers that satisfy the system of equations

\begin{align*} a + b &= -3, \\ ab + bc + ca &= -4, \\ abc + bcd + cda + dab &= 14, \\ abcd &= 30. \end{align*}

There exist relatively prime positive integers 

$m$

 and 

$n$

 such that

\[a^2 + b^2 + c^2 + d^2 = \frac{m}{n}.\]

Find 

$m + n$

.

Q8 2015 IMO Problems/Problem 2

Determine all triples of positive integers $(a,b,c)$ such that each of the numbers\[ab-c,\; bc-a,\; ca-b\]is a power of 2.

(A power of 2 is an integer of the form  where  is a non-negative integer ).

Q1 276

Q2

$(21, 2, 1)$

 and 

$(1, 22, 1)$

Q3 12

Q4 160

Q5 15

Q6 200

Q7 145

Q8 We obtain $(a,b,c)=(3,5,7)$ as the only solution with $3 \leq a < b \leq c$.

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One response to “代数因式分解 + 质数整数因数分析 Prime Integer Divisor[c in progress]”

  1. radmin Avatar
    radmin

    C done Q1 Q2 Q3

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