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?
+2667 1) \({2 \over 9} + {1 \over 7} = 0. \overline{365079}\)
I'll let you work it from here.
+2667 2) There are 6 numbers:
125
152
215
251
512
521
Just find the prime numbers from here...
+2667 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 \(2^2 \times 3^2 \times 5 = \color{brown}\boxed{60}\)
+2667 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.
