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1. What is the 20th digit in the decimal expansion for the sum of 2/9 and 1/7?

2. The digits 1, 2, and 5 can be scrambled to form three-digit numbers. How many of the numbers are prime?

3. The integer x has 12 positive factors. The numbers 12 and 15 are factors of x. What is x?

4. Cammie has some pennies, nickels, dimes, and quarters. What is the least number of coins that she can use to make 93 cents?

Sep 13, 2022

#2
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1) $${2 \over 9} + {1 \over 7} = 0. \overline{365079}$$

I'll let you work it from here.

Sep 15, 2022
#3
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2) There are 6 numbers:

125

152

215

251

512

521

Just find the prime numbers from here...

Sep 15, 2022
#4
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3) $$12 = 2^2 \times 3$$$$15 = 3 \times 5$$

This tells us that the number will have 3 prime factors (2, 3, and 5).

The key here is that $$2^x \times 3^y \times 5^z$$ has $$(x+1)(y+1)(z+1)$$

The only 3 numbers that multiply to 12 are 2, 2, and 3.

So, z = 1, y = 1, and x = 2 meaning the number is $$2^2 \times 3^2 \times 5 = \color{brown}\boxed{60}$$

Sep 15, 2022
#5
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4) First, maximize the quarters, then maximize the dimes with what's left, then repeat for the nickels and pennies.

So, use 3 quarters for 75 cents

Then, use 1 dime to get to 85

Then, use 1 nickel to get to 90

Then, use 3 pennies to get to 93

So $$3 + 3 + 1 + 1 = \color{brown}\boxed{8}$$ coins.

Sep 15, 2022