A Multiplication Based Logic Puzzle

One of the things I really like about these Challenge puzzles is that you have to use logic. Guessing and checking just won’t work. Go ahead and give this puzzle a try!

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

1 – 0 + 3 – 4 = 0, so 1034 can be evenly divided by 11.

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1033 and Level 3

To solve a level 3 puzzle begin with 80, the clue at the very top of the puzzle. Clue 48 goes with it. What are the factor pairs of those numbers in which both factors are between 1 and 12 inclusive? 80 can be 8×10, and 48 can be 4×12 or 6×8. What is the only number that listed for both 80 and 48? Put that number in the top row over the 80. Put the corresponding factors where they go starting at the top of the first column.

Work down that first column cell by cell finding factors and writing them as you go. Three of the factors have been highlighted because you have to at least look at the 55 and the 5 to deal with the 20 in the puzzle. Have fun!

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

Here are a few facts about the number 1033:

It is a twin prime with 1031.

32² + 3² = 1033, so it is the hypotenuse of a Pythagorean triple:
192-1015-1033 calculated from 2(32)(3), 32² – 3², 32² + 3²

1033 is a palindrome in two other bases:
It’s 616 in BASE 13 because 6(13²) + 1(13) + 6(1) = 1033
1J1 in BASE 24 (J is 19 base 10) because 24² + 19(24) + 1 = 1033

8¹ + 8º + 8³ + 8³ = 1033 Thanks to Stetson.edu for that fun fact!

  • 1033 is a prime number.
  • Prime factorization: 1033 is prime.
  • The exponent of prime number 1033 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1033 has exactly 2 factors.
  • Factors of 1033: 1, 1033
  • Factor pairs: 1033 = 1 × 1033
  • 1033 has no square factors that allow its square root to be simplified. √1033 ≈ 32.1403

How do we know that 1033 is a prime number? If 1033 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1033 ≈ 32.1. Since 1033 cannot be divided evenly by 2, 3, 5, 7, 3, 13, 17, 19, 23, 29 or 31, we know that 1033 is a prime number.

Here’s another way we know that 1033 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 32² + 3² = 1033 with 32 and 3 having no common prime factors, 1033 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1033 ≈ 32.1. Since 1033 is not divisible by 5, 13, 17, or 29, we know that 1033 is a prime number.

 

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

You might think that a day lasts 24 hours, but strategic use of the international date line can actually make a single day last 48 hours!

How will you spend the 48 hour day that will be 7 March 2018?

Colleen Young encourages you and your class to register for and participate in World Maths Day 2018 held that day. She shares the necessary links as well as several tips on how to prepare.

One way to prepare now is playing multiplication games like the level 1 puzzle below. The puzzle is just a multiplication table but the factors are missing and only a few of the products are given, and they aren’t in the order you would normally expect. Can you figure out where the factors from 1 to 12 belong in both the first column and the top row of the puzzle?

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

If this puzzle is too easy for you, Then it is time to move on to a level 2 or higher puzzle. You can find one in the link above and plenty others here at findthefactors.com.

Now I’d like to tell you some things that I’ve learned about the number 1031:

1031 and 1033 are twin primes.

1031 is a palindrome in a couple of bases:
It’s 858 in BASE 11 because 8(121) + 5(11) + 8(1) = 1031 and
it’s 272 in BASE 21 because 2(441) + 7(21) + 2(1) = 1031

  • 1031 is a prime number.
  • Prime factorization: 1031 is prime.
  • The exponent of prime number 1031 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1031 has exactly 2 factors.
  • Factors of 1031: 1, 1031
  • Factor pairs: 1031 = 1 × 1031
  • 1031 has no square factors that allow its square root to be simplified. √1031 ≈ 32.10919

How do we know that 1031 is a prime number? If 1031 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1031 ≈ 32.1. Since 1031 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1031 is a prime number.

Love can often be like a puzzle. When cupid’s arrow hits its mark, at first everything might seem to fall into place, but before long, love starts getting complicated and has to be figured out.

That’s the way this Cupid’s Arrow puzzle is, too. It’s easy to find the logic to start it, but then the logic will be more difficult to see. May you be able to figure out this puzzle as well as the important relationships in your life!

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

What can I tell you about the number 1030?

It’s the sum of two consecutive prime numbers:
509 + 521 = 1030

It’s the hypotenuse of a Pythagorean triple:
618-824-1030 which is (3-4-5) times 206

It’s palindrome 1102011 in BASE 3 because 3⁶ + 3⁵ + 2(3³) + 3¹ + 3⁰ = 1030

  • 1030 is a composite number.
  • Prime factorization: 1030 = 2 × 5 × 103
  • 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 1030 has exactly 8 factors.
  • Factors of 1030: 1, 2, 5, 10, 103, 206, 515, 1030
  • Factor pairs: 1030 = 1 × 1030, 2 × 515, 5 × 206, or 10 × 103
  • 1030 has no square factors that allow its square root to be simplified. √1030 ≈ 32.09361

 

A dozen roses can be pretty pricey around Valentine’s Day, but at least one website asserts that a single rose can make just as big a statement and just as big an impact. Today’s mystery level puzzle looks like a single rose. I hope you will enjoy its beauty even if its thorns are prickly.

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

Here are a few facts about the number 1029:

It’s easy to see that 1029 can be evenly divided by 3 because 1 + 0 + 2 + 9 = 12, a number divisible by 3.

It’s not quite as easy to tell that it can be evenly divided by 7:
It is because 102 – 2(9) = 102 – 18 = 84, a number divisible by 7.

I like the way 1029 looks when it is written in some other bases:
It’s 4433 in BASE 6 because 4(6³) + 4(6²) + 3(6¹) + 3(6⁰) = 4(216 + 36) + 3(6 + 1) = 1029,
3000 in BASE 7 because 3(7³) = 3(343) = 1029,
399 in BASE 17 because 3(17²) + 9(17) + 9(1) = 3(289 + 51 + 3) = 3(343) = 1029
333 in BASE 18 because 3(18² + 18 + 1) = 3(343) = 1029

  • 1029 is a composite number.
  • Prime factorization: 1029 = 3 × 7 × 7 × 7, which can be written 1029 = 3 × 7³
  • The exponents in the prime factorization are 1 and 3. Adding one to each and multiplying we get (1 + 1)(3 + 1) = 2 × 4 = 8. Therefore 1029 has exactly 8 factors.
  • Factors of 1029: 1, 3, 7, 21, 49, 147, 343, 1029
  • Factor pairs: 1029 = 1 × 1029, 3 × 343, 7 × 147, or 21 × 49
  • Taking the factor pair with the largest square number factor, we get √1029 = (√49)(√21) = 7√21 ≈ 32.07803

 

Valentine’s Day is almost here so I’ve made three Valentine related puzzles this week. I’ve labeled all of three of them Mystery Level because you might find some of them to be difficult. Use logic and an ordinary 12 × 12 multiplication table. I promise that each one of them can be solved, and I hope that you LOVE working on them! There will be some easier puzzles later on in the week.

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

Let me tell you a little about the number 1028:

32² + 2² = 1028 so 1028 is the hypotenuse of a Pythagorean triple:
128-1020-1028 which is 4 times (32-255-257) and can be calculated from 2(32)(2), 32² – 2², 32² + 2²

1028 is a palindrome when it is written in a couple of different bases:
404 in BASE 16 because 4(16²) + 4(1) = 4(257) = 1028
2G2 in BASE 19 (G is 16 base 10) because 2(19²) + 16(19) + 2(1) = 1028

  • 1028 is a composite number.
  • Prime factorization: 1028 = 2 × 2 × 257, which can be written 1028 = 2² × 257
  • The exponents in the prime factorization are 2 and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1) = 3 × 2  = 6. Therefore 1028 has exactly 6 factors.
  • Factors of 1028: 1, 2, 4, 257, 514, 1028
  • Factor pairs: 1028 = 1 × 1028, 2 × 514, or 4 × 257
  • Taking the factor pair with the largest square number factor, we get √1028 = (√4)(√257) = 2√257 ≈ 32.062439

 

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