Which pins must be knocked over to score exactly 100 points? (Hint: There are three!)

*Answer*: 13, 39, and 48.

Which pins must be knocked over to score exactly 100 points? (Hint: There are three!)

*Answer*: 13, 39, and 48.

96 = 2^5 x 3^1

54 x 18 = 972

As all the numbers in the 990s repeat a 9 and 989 and 988 also have repeating digits, the first number to check is 987. Count down from there checking each number for its factors.

Q: What is the difference between the sum of the first 2023 even counting mumbers and the sum of the first 2023 odd counting numbers?

A: 2023

If we “solve a simpler problem” (like Polya recommended), we can try the first 10 even counting numbers and the first 10 odd counting numbers.

2 + 4 + 6 + 8 + 10 + 12 + 14 + 16 + 18 + 20

1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19

Notice that each of the even numbers in the top row is one more than its corresponding odd number below it. Therefore, the difference between the sum of these rows of numbers is 10.

If we now expand that idea to the first 2023 even numbers vs the first 2023 odd numbers, each of the even numbers is 1 more than its corresponding odd number. Therefore, the difference is 2023.

Click here to see the AMC 12 that inspired this problem

Q: Without a calculator or the Internet, find the prime factors of 2023.

A: Try prime numbers.

First: 2 the number is not even so 2 is not a factor.

For 3, we have to check the sum of the digits which total to 7 which is not divisible by 3, therefore 2023 is not divisible by 3.

For 5, the number doesn’t end in 5 or 0 so not divisible by 5.

For 7, think of a number close that is divisible by 7 — 2100 is 300 x 7.

2023 is 77 less than 2100. Since 77 is also divisible by 7, 2023 is a multiple of 7. It is 300 x 7 – 11 x 7 which is 289 x 7.

289 happens to be a perfect square as it is 17 x 17.

Since 17 is prime, it cannot be factored any further.

So the prime factors of 2023 are: 7 x 17 x 17.