A Multiplication Based Logic Puzzle

Archive for the ‘Level 6 Puzzle’ Category

958 and Level 6

This puzzle is a multiplication table. You don’t have to be fast to solve it, but you do have to think. There is only one solution. The ten clues given are sufficient to find the places to put the factors 1 to 10 in the first column and the top row. Good luck!

Print the puzzles or type the solution in this excel file: 10-factors-951-958

958 is the sum of the 22 prime numbers from 5 to 89.

958 is also 141 in BASE 29 because 1(29²) + 4(29¹) + 1(29⁰) = 958

  • 958 is a composite number.
  • Prime factorization: 958 = 2 × 479
  • The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 958 has exactly 4 factors.
  • Factors of 958: 1, 2, 479, 958
  • Factor pairs: 958 = 1 × 958 or 2 × 479
  • 958 has no square factors that allow its square root to be simplified. √958 ≈ 30.951575

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950 and Level 6

Today’s puzzle is a level 6, so it isn’t for beginners. It is trickier than the easier levels, but don’t let that stop you from giving it a try. If you use guess and check, you will likely become frustrated, but you can solve this puzzle IF you use logic before you write each factor.

Print the puzzles or type the solution in this excel file: 12 factors 942-950

950 is the hypotenuse of two Pythagorean triples:
266-912-950 which is (7-24-25) times 38
570-760-950 which is (3-4-5) times 190

950 is 2525 in BASE 7 because 2(7³) + 5(7²) + 2(7¹) + 5(7⁰) = 950

  • 950 is a composite number.
  • Prime factorization: 950 = 2 × 5 × 5 × 19, which can be written 950 = 2 × 5² × 19
  • The exponents in the prime factorization are 2, 1, and 1. Adding one to each and multiplying we get (1 + 1)(2 + 1)(1 + 1) = 2 × 3 × 2 = 12. Therefore 950 has exactly 12 factors.
  • Factors of 950: 1, 2, 5, 10, 19, 25, 38, 50, 95, 190, 475, 950
  • Factor pairs: 950 = 1 × 950, 2 × 475, 5 × 190, 10 × 95, 19 × 50, or 25 × 38,
  • Taking the factor pair with the largest square number factor, we get √950 = (√25)(√38) = 5√38 ≈ 30.82207

 

941 and Level 6

Today daylight savings time ends in the United States. That gives you an extra hour to sleep or do whatever you want. Perhaps you could spend part of your extra hour solving today’s puzzle:

Print the puzzles or type the solution on this excel file: 10-factors-932-941

941 is a prime number that is also the sum of consecutive prime numbers two different ways:
179 + 181 + 191 + 193 + 197 = 941; that’s five consecutive primes
311 + 313 + 317 = 941; that’s three consecutive primes

29² + 10² = 941 That means 941 is the hypotenuse of a Pythagorean triple:
580-741-941 which can be calculated from 2(29)(10), 29² – 10², 29² + 10²

941 is also palindrome 575 in BASE 13 because 5(13²) + 7(13¹) + 5(13⁰) = 941

  • 941 is a prime number.
  • Prime factorization: 941 is prime.
  • The exponent of prime number 941 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 941 has exactly 2 factors.
  • Factors of 941: 1, 941
  • Factor pairs: 941 = 1 × 941
  • 941 has no square factors that allow its square root to be simplified. √941 ≈ 30.6757233

How do we know that 941 is a prime number? If 941 were not a prime number, then it would be divisible by at least one prime number less than or equal to √941 ≈ 30.7. Since 941 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, or 29, we know that 941 is a prime number.

Here’s another way we know that 941 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 29² + 10² = 941 with 29 and 10 having no common prime factors, 941 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √941 ≈ 30.7. Since 941 is not divisible by 5, 13, 17, or 29, we know that 941 is a prime number.

 

930 Witch’s Broom

This witch’s broom is a puzzle based on the multiplication table. The factors from 1 to 12 are out of order in the first column and the top row, but they all belong somewhere in each place. Use logic to figure out where those factors go. It’s a level 6, so it will be a little tricky, but not as tricky as it sometimes is. Solve it, and you might feel like flying!

Print the puzzles or type the solution on this excel file: 12 factors 923-931

What is special about the number 930?

463 + 467 = 930, so it is the sum of consecutive prime numbers.

30 × 31 = 930 The factors in that factor pair are consecutive so 930 is the sum of the first 30 EVEN numbers:
2 + 4 + 6 + 8 + . . . + 54 + 56 + 58 + 60 = 930

930 is the hypotenuse of a Pythagorean triple:
558-744-930, which is (3-4-5) times 186.

Here’s 930 in a few other bases:
Palindrome 656 in BASE 12, because 6(144) + 5(12) + 6(1) = 930
110 in BASE 30, because 1(30²) + 1(30) + 0(1) = 30(31) = 930
U0 in BASE 31 (U is 30 base 10), because 30(31) + 0(1) = 930

  • 930 is a composite number.
  • Prime factorization: 930 = 2 × 3 × 5 × 31
  • The exponents in the prime factorization are 1, 1, 1, and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 × 2 = 16. Therefore 930 has exactly 16 factors.
  • Factors of 930: 1, 2, 3, 5, 6, 10, 15, 30, 31, 62, 93, 155, 186, 310, 465, 930
  • Factor pairs: 930 = 1 × 930, 2 × 465, 3 × 310, 5 × 186, 6 × 155, 10 × 93, 15 × 62, or 30 × 31
  • 930 has no square factors that allow its square root to be simplified. √930 ≈ 30.495901.

922 Boo!

Have you ever cut holes in a sheet, put it over your head, and jumped out in front of people as you hollered, “Boo!”? Today’s puzzle is made to look like a ghost. It’s a level 6, but don’t let that spook you! Attack the puzzle using logic, and after you solve it, you can claim to be a ghost-buster!

Print the puzzles or type the solution on this excel file: 10-factors-914-922

I think my ghost is cute. Maybe not as cute as …

https://platform.twitter.com/widgets.js

When 922 floats around in a different base, you may think you’re seeing an apparition:

922 becomes 1234 in BASE 9 because 1(9³) + 2(9²) + 3(9¹) + 4(9º) = 922.
922 becomes palindrome 262 in BASE 20

922 is also the sum of the 18 prime numbers from 17 to 89.

922 = 29² + 9², so 922 is the hypotenuse of a Pythagorean triple:
522-760-922, which is the same as 2(29)(9), 29² – 9², 29² + 9².
That Pythagorean triple is also 2 times (261-380-461).

  • 922 is a composite number.
  • Prime factorization: 922 = 2 × 461
  • The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 922 has exactly 4 factors.
  • Factors of 922: 1, 2, 461, 922
  • Factor pairs: 922 = 1 × 922 or 2 × 461
  • 922 has no square factors that allow its square root to be simplified. √922 ≈ 30.3644529

 

 

912 and Level 6

912 is the sum of the ten prime numbers from 71 to 109.

It is also the sum of these four consecutive primes:

  • 223 + 227 + 229 + 233 = 912

912 is 192 in BASE 26 because 1(26²) + 9(26) + 2(1) = 912.

Print the puzzles or type the solution on this excel file: 12 factors 905-913

  • 912 is a composite number.
  • Prime factorization: 912 = 2 × 2 × 2 × 2 × 3 × 19, which can be written 912 = 2⁴ × 3 × 19
  • The exponents in the prime factorization are 4, 1 and 1. Adding one to each and multiplying we get (4 + 1)(1 + 1)(1 + 1) = 5 × 2 × 2 = 20. Therefore 912 has exactly 20 factors.
  • Factors of 912: 1, 2, 3, 4, 6, 8, 12, 16, 19, 24, 38, 48, 57, 76, 114, 152, 228, 304, 456, 912
  • Factor pairs: 912 = 1 × 912, 2 × 456, 3 × 304, 4 × 228, 6 × 152, 8 × 114, 12 × 76, 16 × 57, 19 × 48 or 24 × 38
  • Taking the factor pair with the largest square number factor, we get √912 = (√16)(√57) = 4√57 ≈ 30.1993377.

904 and Level 6

30² + 2² = 904

That means 904 is the hypotenuse of a Pythagorean triple:

  • 120-896-904 which is 8 times (15-112-113)

Print the puzzles or type the solution on this excel file: 10-factors-897-904

  • 904 is a composite number.
  • Prime factorization: 904 = 2 × 2 × 2 × 113, which can be written 904 = 2³ × 113
  • The exponents in the prime factorization are 3 and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1) = 4 × 2 = 8. Therefore 904 has exactly 8 factors.
  • Factors of 904: 1, 2, 4, 8, 113, 226, 452, 904
  • Factor pairs: 904 = 1 × 904, 2 × 452, 4 × 226, or 8 × 113
  • Taking the factor pair with the largest square number factor, we get √904 = (√4)(√226) = 2√226 ≈ 30.06659

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