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AMC10 8 wk prep w20-week 1 assignment

 
 
Picture of Leo Zeng
AMC10 8 wk prep w20-week 1 assignment
by Leo Zeng - Tuesday, December 1, 2020, 9:54 PM
 

I don't get questions 3,9, and 10. I'm stuck. Can I get help?

 
Picture of David Reynoso
Re: AMC10 8 wk prep w20-week 1 assignment
by David Reynoso - Wednesday, December 2, 2020, 2:39 PM
 

Here are some hints to get you started:

3: Write $\dfrac{a}{b} + \dfrac{14b}{9a}$ as a single fraction. What do you need to ask of $a$ and $b$ so that the resulting fraction is actually an integer?

9:  Start by looking at the ones digits of $N$ in base $10$, $5$, and $6$. For example, if the last digit of $N$ is base $10$ is $7$, the last digit of $N$ in base $5$ is $2$, and the last digit of $2N$ in base $10$ is $4$, thus for $S$ to have the same last digit as $2N$, the last digit of $N$ in base $6$ must be $4 - 2 = 2$.  (Note a digit in a number expressed in base $5$ is at most $4$, and in base $6$ is at most $5$, so there will not be any carryover when these are added as if they were numbers in base $10$.)

10: Find a pattern for the remainders of powers of $3$ when divided by $40$.

Picture of Duy Duong
Re: AMC10 8 wk prep w20-week 1 assignment
by Duy Duong - Friday, December 4, 2020, 6:49 PM
 

For number 10 as well, used what you know from problem 2