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1. A segment has endpoints P(1,1) and Q(x,y). The coordinates of the midpoint of segment PQ are positive integers with a producr of 36. What is the maximum possible value of x?

2. Given that 591,3d8 is divisible by 12, what is the sum of all of the digits that could replace d?

3. The integer 6 has four positive divisors: 1, 2, 3 and 6. What is the smallest positivie integer with exactly five positive divisors?

4. What is the units digit of 2013^2013?

5. What is the sum of all of the two-digit primes that are greater than 12 but less than 99 and are still prime when their two digits are interchanged?

 Sep 15, 2022
 #1
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1) Maximize the x coordinate of the midpoint, and go from there. 

 Sep 15, 2022
 #6
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would the answer be 11??

Guest Sep 16, 2022
 #2
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2) If it is divisible by 12, it must be divisible by 4 and 3. The only cases where it is divisible by 4 are 2, 4, 6, and 8. 

 

If it is divisible by 3, d must be 1 more than a multiple of 3. The only case that works is 4, so the sum is \(\color{brown}\boxed{4}\)

 Sep 15, 2022
 #3
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3) If a number has an odd number of divisors, it must be a square number. 

 

Now, just try each square number until you find the smallest one with 5 factors. 

 Sep 15, 2022
 #7
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So the correct answer is 16?

Guest Sep 16, 2022
 #4
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4) Track the last digit: 3, 9, 7, 1, 3, 9, 7, 1, 3 ...

 

Notice a pattern?

 Sep 15, 2022
 #5
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5) If a 2-digit number is prime, it must end in 1, 3, 7, or 9. 

 

This means that the only numbers that satisfy the condition will contain 2 of the 4 numbers listed above. 

 

Try each one out and find the ones that work.

 Sep 15, 2022

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