A Multiplication Based Logic Puzzle

Archive for the ‘Level 2 Puzzle’ Category

1032 How Many Twelves Are in a 12×12 Times Table?

How many 12’s are in a standard 12×12 times table? There are five 12’s in the puzzle below. Six, if you count the 12 in the title. Is that too many, just the right number, or are there even more?

This is only a level 2 puzzle so it won’t be difficult to solve. . . unless I’ve put in too many 12’s!

Hmm. . .Try solving the puzzle, then fill in the rest of the multiplication table. Then you will know for sure how many 12’s SHOULD be in the table.

Print the puzzles or type the solution in this excel file: 12 factors 1028-1034

The number 1032 is divisible by 12. Here are a few more facts about that number:

1032 is made with a zero and three consecutive numbers so it is divisible by 3.

The last two digits of 1032 are 32 so 1032 can be evenly divided by 4.

Since 32 is divisible by 8 and preceded by an even zero in 1032, our number is also divisible by 8.

As you will soon see, 1032 is divisible by even more numbers than those listed above. Here are three of its factor trees:

1032 looks interesting in a couple of different bases:
It’s 4440 in BASE 6 because 4(6³ + 6² + 6¹) = 4(258) = 1032, and
it’s palindrome 3003 in BASE 7 because 3(7³ + 7⁰) = 3(344) = 1032.

  • 1032 is a composite number.
  • Prime factorization: 1032 = 2 × 2 × 2 × 3 × 43, which can be written 1032 = 2³ × 3 × 43
  • The exponents in the prime factorization are 3, 1, and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1)(1 + 1) = 4 × 2 × 2 = 16. Therefore 1032 has exactly 16 factors.
  • Factors of 1032: 1, 2, 3, 4, 6, 8, 12, 24, 43, 86, 129, 172, 258, 344, 516, 1032
  • Factor pairs: 1032 = 1 × 1032, 2 × 516, 3 × 344, 4 × 258, 6 × 172, 8 × 129, 12 × 86, or 24 × 43
  • Taking the factor pair with the largest square number factor, we get √1032 = (√4)(√258) = 2√258 ≈ 32.12476

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Level 2 and Simplifying √1014

Have you memorized a basic multiplication table? If you have, then you can solve this puzzle. The numbers being multiplied together aren’t where they are in a regular multiplication table, but you can still easily figure out where they need to go. There is only one solution. I bet you can find it!

Print the puzzles or type the solution in this excel file: 12 factors 1012-1018

If I wanted to find √1014, I would first check to see if it were divisible by 4 or by 9 because most numbers whose square roots can be simplified are divisible by 4 or by 9 or both.
1014 isn’t divisible by 4 because 14 isn’t divisible by 4.
It isn’t divisible by 9 because 1 + 0 + 1 + 4 = 6, and 6 is not divisible by 9.
However, it is divisible by both 2 and 3 and thus also by 6. Since most people are less likely to make a mistake dividing by 6 in ONE step instead of two, I would make a little division cake and do that division first:

Recognizing that 169 is a perfect square, I would then take the square root of everything on the outside of my little cake. (√6)(√169) = 13√6

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Here are some facts about the number 1014:

Because 13² is one of its factors, 1014 is the hypotenuse of two Pythagorean triples:
714-720-1014 which is 6 times (119-120-169),
390-936-1014 which is (5-12-13) times 78

1014 looks interesting when written in some other bases:
It’s 600 in BASE 13 because 6(13²) = 6 (169) = 1014,
and 222 in BASE 22 because 2(22²) + 2(22) + 2(1) = 2(484 + 22 + 1) = 2(507) = 1014

  • 1014 is a composite number.
  • Prime factorization: 1014 = 2 × 3 × 13 × 13, which can be written 1014 = 2 × 3 × 13²
  • The exponents in the prime factorization are 1, 1, and 2. Adding one to each and multiplying we get (1 + 1)(1 + 1)(2 + 1) = 2 × 2 × 3 = 12. Therefore 1014 has exactly 12 factors.
  • Factors of 1014: 1, 2, 3, 6, 13, 26, 39, 78, 169, 338, 507, 1014
  • Factor pairs: 1014 = 1 × 1014, 2 × 507, 3 × 338, 6 × 169, 13 × 78, or 26 × 39,
  • Taking the factor pair with the largest square number factor, we get √1014 = (√169)(√6) = 13√6 ≈ 31.84337

1003 and Level 2

There is only one way to write the numbers 1 to 10 in both the first column and the top row so that you create a multiplication table and the clues in the puzzle belong where they are. Can you find that way? It easy difficult. Give it a try!

Print the puzzles or type the solution in this excel file: 10-factors-1002-1011

1003 is the hypotenuse of a Pythagorean triple:
472∗885∗1003 which is (8-15-17) times 59

1003 is also palindrome 1101011 in BASE 3
because 3⁶ + 3⁵ + 0(3⁴) + 3³ + 0(3²) + 3¹ + 3⁰ = 1003

  • 1003 is a composite number.
  • Prime factorization: 1003 = 17 × 59
  • 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 1003 has exactly 4 factors.
  • Factors of 1003: 1, 17, 59, 1003
  • Factor pairs: 1003 = 1 × 1003 or 17 × 59
  • 1003 has no square factors that allow its square root to be simplified. √1003 ≈ 31.670175

995 and Level 2

If you know the only number that will divide evenly into 2, 11, 10, 4, 1, and 9, then you can easily solve this level 2 puzzle. There are even elementary aged kids that you know who can solve it. I’m sure that together you can fill in every cell in the entire puzzle. Don’t be afraid. Just do it.

Print the puzzles or type the solution in this excel file: 12 factors 993-1001

Here’s some stuff that you probably didn’t know about the number 995:

995 has only two factor pairs, and they both use the exact same digits (1, 9, 9, 5):
1 × 995 =  199 × 5

It is the hypotenuse of a Pythagorean triple:
597-796-995 which is (3-4-5) times 199

It is palindrome 3E3 in BASE 16 (E is 14 base 10) because 3(16²) + 14(16) + 3(1) = 995

  • 995 is a composite number.
  • Prime factorization: 995 = 5 × 199
  • 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 995 has exactly 4 factors.
  • Factors of 995: 1, 5, 199, 995
  • Factor pairs: 995 = 1 × 995 or 5 × 199
  • 995 has no square factors that allow its square root to be simplified. √995 ≈ 31.54362

 

 

987 Christmas Star

Today’s puzzle is a lovely Christmas star whose golden beams shine throughout the dark night. Solving this puzzle could also enlighten your mind.

Print the puzzles or type the solution in this excel file: 10-factors-986-992

987 is made from three consecutive numbers so it is divisible by 3.

It is also the sixteen number in the Fibonacci sequence:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, . . .

987 is a palindrome when written in base 11 or base 29:
818 in BASE 11 because 8(121) + 1(11) + 8(1) = 987
151 in BASE 29 because 1(29²) + 5(29) + 1(1) = 987

  • 987 is a composite number.
  • Prime factorization: 987 = 3 × 7 × 47
  • The exponents in the prime factorization are 1, 1, and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 = 8. Therefore 987 has exactly 8 factors.
  • Factors of 987: 1, 3, 7, 21, 47, 141, 329, 987
  • Factor pairs: 987 = 1 × 987, 3 × 329, 7 × 141, or 21 × 47
  • 987 has no square factors that allow its square root to be simplified. √987 ≈ 31.416556

 

979 Was virgács in your shoes this morning?

Today was Saint Nicholas day in much of Europe. Children woke up and looked in their shoes that they had carefully laid out the night before. They love to find their favorite candies letting them know they’ve been good this last year. In Hungary, where everyone’s behavior is considered to be a mixture of both good and bad, children also found some virgács in their shoes, letting them know they were also naughty some of the time. Today’s puzzle looks a little like virgács.

Print the puzzles or type the solution in this excel file: 12 factors 978-985

1⁴ + 2⁴ + 3⁴ + 4⁴ + 5⁴ = 979

979 is the sum of the eleven prime numbers from 67 to 109.

979 is a palindrome in base 10 and in two other bases:
454 in BASE 15
3D3 in BASE 16 (D is 13 base 10)

979 is the hypotenuse of a Pythagorean triple:
429-880-979 which is 11 times (39-80-89)

  • 979 is a composite number.
  • Prime factorization: 979 = 11 × 89
  • 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 979 has exactly 4 factors.
  • Factors of 979: 1, 11, 89, 979
  • Factor pairs: 979 = 1 × 979 or 11 × 89
  • 979 has no square factors that allow its square root to be simplified. √979 ≈ 31.288976

971 and Level 2

Some of the clues in today’s puzzles are perfect squares. Some aren’t. Can you figure out which are which and put their factors in the right places in the first column and the top row?

Print the puzzles or type the solution in this excel file: 10-factors-968-977

971 is palindrome 2D2 in BASE 19 (D is 13 base 10)
because 2(19²) + 13(19) + 2(1) = 971

  • 971 is a prime number.
  • Prime factorization: 971 is prime.
  • The exponent of prime number 971 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 971 has exactly 2 factors.
  • Factors of 971: 1, 971
  • Factor pairs: 971 = 1 × 971
  • 971 has no square factors that allow its square root to be simplified. √971 ≈ 31.16087

How do we know that 971 is a prime number? If 971 were not a prime number, then it would be divisible by at least one prime number less than or equal to √971 ≈ 31.2. Since 971 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 971 is a prime number.

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