1243 and Level 1

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Here’s a great puzzle to help students figure out some division facts. That’s what they will have to do to find the factors from 1 to 10. Once they find those factors, they can complete the puzzle like it is a multiplication table.

Print the puzzles or type the solution in this excel file: 10-factors-1242-1250

Now I’ll share a few facts about the number 1243:

  • 1243 is a composite number.
  • Prime factorization: 1243 = 11 × 113
  • 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 1243 has exactly 4 factors.
  • Factors of 1243: 1, 11, 113, 1243
  • Factor pairs: 1243 = 1 × 1243 or 11 × 113
  • 1243 has no square factors that allow its square root to be simplified. √1243 ≈ 35.25621

1243 is also the hypotenuse of a Pythagorean triple:
165-1232-1243 which is 11 times (15-112-113)

 

1242 is a Decagonal Number

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If you had 1242 tiny little squares you could arrange them into a decagon, just as I did for the graphic below.

18 is a factor of 1242. Since 18(4·18-3) = 18(69) = 1242, it is the 18th decagonal number.

Here are a few more facts about the number 1242:

  • 1242 is a composite number.
  • Prime factorization: 1242 = 2 × 3 × 3 × 3 × 23, which can be written 1242 = 2 × 3³ × 23
  • The exponents in the prime factorization are 1, 3, and 1. Adding one to each and multiplying we get (1 + 1)(3 + 1)(1 + 1) = 2 × 2 × 2 = 16. Therefore 1242 has exactly 16 factors.
  • Factors of 1242: 1, 2, 3, 6, 9, 18, 23, 27, 46, 54, 69, 138, 207, 414, 621, 1242
  • Factor pairs: 1242 = 1 × 1242, 2 × 621, 3 × 414, 6 × 207, 9 × 138, 18 × 69, 23 × 54, or 27 × 46
  • Taking the factor pair with the largest square number factor, we get √1242 = (√9)(√138) = 3√138 ≈ 35.24202

1242 is the sum of consecutive prime numbers two different ways:
It is the sum of the eighteen prime numbers from 31 to 107, and
it is also the sum of the sixteen prime numbers from 43 to 109.

1241 Mystery Level

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Sure, you know many of the factors of the clues in this puzzle, but don’t write the first one that pops in your head. You might be lucky, but you also might have to do a lot of erasing if you do. Study all the clues and let logic tell you where to put each factor.

Print the puzzles or type the solution in this excel file: 12 factors 1232-1241

Since this is the 1241st post, I’ll write a little about the number 1241:

  • 1241 is a composite number.
  • Prime factorization: 1241 = 17 × 73
  • 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 1241 has exactly 4 factors.
  • Factors of 1241: 1, 17, 73, 1241
  • Factor pairs: 1241 = 1 × 1241 or 17 × 73
  • 1241 has no square factors that allow its square root to be simplified. √1241 ≈ 35.22783

1241 is the sum of two squares two different ways:
29² + 20² = 1241
35² + 4² =1241

1241 is the hypotenuse of FOUR Pythagorean triples:
816-935-1241 which is 17 times (48-55-73)
584-1095-1241 which is (8-15-17) times 73
280-1209-1241 calculated from 2(35)(4), 35² – 4², 35² + 4²
441-1160-1241 calculated from 29² – 20², 2(29)(20), 29² + 20²

1240 is a Square Pyramidal Number

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1240 is the 15th square pyramidal number because
1² + 2² + 3² + 4² + 5² + 6² + 7² + 8² + 9² + 10² + 11² + 12² + 13² + 14² + 15² = 1240

We can know that 1240 is the 15th square pyramidal number because
15(15 + 1)(2·15 + 1)/6
= 15(16)(31)/6
= (5)(8)(31)
= (40)(31)
= 1240

Here are some more facts about the number 1240:

  • 1240 is a composite number.
  • Prime factorization: 1240 = 2 × 2 × 2 × 5 × 31, which can be written 1240 = 2³ × 5 × 31
  • 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 1240 has exactly 16 factors.
  • Factors of 1240: 1, 2, 4, 5, 8, 10, 20, 31, 40, 62, 124, 155, 248, 310, 620, 1240
  • Factor pairs: 1240 = 1 × 1240, 2 × 620, 4 × 310, 5 × 248, 8 × 155, 10 × 124, 20 × 62, or 31 × 40
  • Taking the factor pair with the largest square number factor, we get √1240 = (√4)(√310) = 2√310 ≈ 35.21363


1240 is the hypotenuse of a Pythagorean triple:
744-992-1240 which is (3-4-5) times 248

1239 Addition and Subtraction Families

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Perhaps in the early years of your education, you were introduced to addition and subtraction families. For example, you might have made a little house out of these four addition and subtraction facts:

3 + 9 = 12
9 + 3 = 12
12 – 9 = 3
12 – 3 = 9

You should have been told then, but weren’t, that there are other members of this addition and subtraction family:

12 = 3 + 9
12 = 9 + 3
3 = 12 – 9
9 = 12 – 3

In fact, those second four addition and subtraction facts may have seemed very strange-looking even years later.

Eventually, you should have been introduced to the whole family of facts involving addition and subtraction and those numbers, but most likely that never happened. Here, the most familiar part of the family can be seen in the dark green part of the house, but the entire rest of the family can also be seen throughout the rest of the house. Some are in the basement and some in the wings of the house, but they all very much belong in this addition and subtraction family home. And it isn’t too difficult to see where every member of the family came from:

If you knew all the members of that family, it would be more natural to accept the members of a family made with variables or numbers mixed with variables:

Instead, many students get very confused when they become teenagers and are introduced to families in which some family members are numbers and some are letters. Likewise, some family members are positive and some are negative.

Going from “a + b = c” to “c – a = b” becomes confusing instead of natural. Required steps involve adding and subtracting the same value from both sides of the equation instead of recalling prior knowledge known since first grade.

I think middle school students might benefit from building an addition and subtraction family house.

Now I would like to share some facts about the number 1239:

  • 1239 is a composite number.
  • Prime factorization: 1239 = 3 × 7 × 59
  • 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 1239 has exactly 8 factors.
  • Factors of 1239: 1, 3, 7, 21, 59, 177, 413, 1239
  • Factor pairs: 1239 = 1 × 1239, 3 × 413, 7 × 177, or 21 × 59
  • 1239 has no square factors that allow its square root to be simplified. √1239 ≈ 35.19943

1239 is divisible by 3 because it is made with three consecutive numbers and a multiple of 3. In this case it isn’t necessary to add the numbers up to see that.

 

1238 and Level 6

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If you use logic, you can figure out the solution to this puzzle. You will have to study all the clues just to know where to start, but I think you’ll find a lot of satisfaction in finding the solution.

Print the puzzles or type the solution in this excel file: 12 factors 1232-1241

Now I’ll share some facts about the number 1238:

  • 1238 is a composite number.
  • Prime factorization: 1238 = 2 × 619
  • 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 1238 has exactly 4 factors.
  • Factors of 1238: 1, 2, 619, 1238
  • Factor pairs: 1238 = 1 × 1238 or 2 × 619
  • 1238 has no square factors that allow its square root to be simplified. √1238 ≈ 35.18522

Because of its prime factors, I know that 1238 is part of only one Pythagorean triple:
1238-383160-383162

1238 is a palindrome in two other bases:
It’s 646 in BASE 14, and
it’s 383 in BASE 19.

1237 and Level 5

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Level 5 puzzles always have at least one set of clues with more than one possible common factor. Still only one of those factors will actually work with all the rest of the clues. Can you use logic to find the factors needed to solve this puzzle?

Print the puzzles or type the solution in this excel file: 12 factors 1232-1241

Let’s look at some facts about the number 1237:

  • 1237 is a prime number.
  • Prime factorization: 1237 is prime.
  • The exponent of prime number 1237 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1237 has exactly 2 factors.
  • Factors of 1237: 1, 1237
  • Factor pairs: 1237 = 1 × 1237
  • 1237 has no square factors that allow its square root to be simplified. √1237 ≈ 35.17101

How do we know that 1237 is a prime number? If 1237 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1237 ≈ 35.1. Since 1237 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1237 is a prime number.

1237 is the sum of two squares:
34² + 9² = 1237

1237 is the hypotenuse of a Pythagorean triple:
612-1075-1237 calculated from 2(34)(9), 34² – 9², 34² + 9²

Here’s another way we know that 1237 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 34² + 9² = 1237 with 34 and 9 having no common prime factors, 1237 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1237 ≈ 35.1. Since 1237 is not divisible by 5, 13, 17, or 29, we know that 1237 is a prime number.

1236 and Level 4

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V is for victory. Can you be victorious solving this puzzle? Write the numbers from 1 to 12 in both the first column and the top row so that the puzzle functions like a multiplication table with the given clues becoming the products of the factors you write. I’m sure you can do it if you stick with it!

Print the puzzles or type the solution in this excel file: 12 factors 1232-1241

Now I’ll tell you some facts about the number 1236:

  • 1236 is a composite number.
  • Prime factorization: 1236 = 2 × 2 × 3 × 103, which can be written 1236 = 2² × 3 × 103
  • The exponents in the prime factorization are 2, 1, and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1)(1 + 1) = 3 × 2 × 2 = 12. Therefore 1236 has exactly 12 factors.
  • Factors of 1236: 1, 2, 3, 4, 6, 12, 103, 206, 309, 412, 618, 1236
  • Factor pairs: 1236 = 1 × 1236, 2 × 618, 3 × 412, 4 × 309, 6 × 206, or 12 × 103
  • Taking the factor pair with the largest square number factor, we get √1236 = (√4)(√309) = 2√309 ≈ 35.15679

1236 is the sum of consecutive prime numbers three rather interesting ways:

  1. It is the sum of the twenty-two prime numbers from 13 to 103.
  2. It is the sum of the eight prime numbers from 137 to 173.
    (137 + 139 + 149 + 151 + 157 + 163 + 167 + 173 = 1236)
  3. It is also the sum of twin primes: 617 + 619 = 1236

1235 and Level 3

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Do you know the greatest common factor of 28 and 35? If you do, then you can solve this puzzle by writing each number from 1 to 12 in both the first column and the top row. Since this is a level 3 puzzle, you can begin with the clues at the top of the puzzle and work your way down cell by cell. Good luck!

Print the puzzles or type the solution in this excel file: 12 factors 1232-1241

Now I’ll share some facts about the number 1235:

  • 1235 is a composite number.
  • Prime factorization: 1235 = 5 × 13 × 19
  • 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 1235 has exactly 8 factors.
  • Factors of 1235: 1, 5, 13, 19, 65, 95, 247, 1235
  • Factor pairs: 1235 = 1 × 1235, 5 × 247, 13 × 95, or 19 × 65
  • 1235 has no square factors that allow its square root to be simplified. √1235 ≈ 35.14257

1235 is the hypotenuse of FOUR Pythagorean triples:
304-1197-1235 which is 19 times (16-63-65)
741-988-1235 which is (3-4-5) times 247
627-1064-1235 which is 19 times (33-56-65)
475-1140-1235 which is (5-12-13) times 95

1234 and Level 2

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This is my 1234th post, so today’s puzzle has been given that number. Whenever I see 12:34 on a clock, I always think about my husband’s Uncle Paul who really liked noticing that time because all possible clock digits are used and the digits are in order. I also like those digits because 12 = 3 × 4.

Print the puzzles or type the solution in this excel file: 12 factors 1232-1241

Here are some facts about the number 1234 some of which might surprise you:

  • 1234 is a composite number.
  • Prime factorization: 1234 = 2 × 617
  • 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 1234 has exactly 4 factors.
  • Factors of 1234: 1, 2, 617, 1234
  • Factor pairs: 1234 = 1 × 1234 or 2 × 617
  • 1234 has no square factors that allow its square root to be simplified. √1234 ≈ 35.12834

1234 is the sum of two squares:
35² + 3² = 1234

1234 is the hypotenuse of a Pythagorean triple:
210-1216-1234 calculated from 2(35)(3), 35² – 3², 35² + 3²
It is also 2 times (105-608-617)