Today’s puzzle looks a little like a wild, but fun? carnival ride. The numbers 36 and 12 went together on the ride. They managed to stay with each other but the ride went so fast, you can see 36 and 12 in two different places at the same time. There’s also poor number 40. You can see it in THREE places at the same time.
Oh my! Can you use logic to find where the numbers 1 to 10 need to go in both the first column and the top row so that this wild ride will behave like a multiplication table? It’s a level 5 so it won’t be easy to find its unique solution. Are you brave enough to try?
Level 5 puzzles aren’t any harder than level 4 puzzles unless I trick you into starting with the common factor of a pair of clues that have more than one possibility. You won’t let me trick you, will you?
1361 has no exponents greater than 1 in its prime factorization, so √1361 cannot be simplified.
The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1361 has exactly 2 factors.
The factors of 1361 are outlined with their factor pair partners in the graphic below.
How do we know that 1361 is a prime number? If 1361 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1361. Since 1361 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1361 is a prime number.
1361 is the first prime number after 1327. That was 34 numbers ago!
1361 is the sum of two squares:
31² + 20² = 1361
1361 is the hypotenuse of a Pythagorean triple:
561-1240-1361 calculated from 31² – 20², 2(31)(20), 31² + 20²
Here’s another way we know that 1361 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 31² + 20² = 1361 with 31 and 20 having no common prime factors, 1361 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1361. Since 1361 is not divisible by 5, 13, 17, or 29, we know that 1361 is a prime number.
When I have a cold or a cough, I often have a piece of hard candy in my mouth. These red cinnamon candies are tasty, but they aren’t very good for soothing throats! Will this red hot cinnamon candy puzzle be too hard for you to solve? It may be a little bit of a challenge, but I’m sure you can solve it if you let logic be your guide the entire time.
The exponent of prime number 1307 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1307 has exactly 2 factors.
Factors of 1307: 1, 1307
Factor pairs: 1307 = 1 × 1307
1307 has no square factors that allow its square root to be simplified. √1307 ≈ 36.15245
How do we know that 1307 is a prime number? If 1307 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1307 ≈ 36.2. Since 1307 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1307 is a prime number.
1307 is palindrome 797 in BASE 13 because 7(13²) + 9(13) + 7(1) = 1307
Only factors from 1 to 12 are allowed in the factor pairs of the clues in these puzzles. Find a row or column with only one allowable common factor to start this puzzle. As you progress, factors for other rows or columns will be eliminated. Keep at it, and you will succeed!