# Find the Factors (ax±b)(cx±d)

I liked making a puzzle using trinomials earlier today. This one will take more skill to solve even though it contains fewer trinomials. Some of the factors will have negative numbers, and the leading coefficients of the trinomials are not 1.

In this puzzle, you can see the number 24 twice. It needs to be factored to solve the puzzle. It might be 3 × 8 or 4 × 6, but it can’t be 1 × 24 or 2 × 12 because for this puzzle ALL of the factors of 24 have to be non-zero integers from -10 to +10. Every factor must appear once in the first column and once in the top row. So if you put 2x + 5 in the top row, you will also have to put 2x + 5 somewhere in the first column as well.

Sometimes all of the terms in the trinomial have a common factor and can, therefore, be factored further, but don’t worry about that right now.

You will have to find all of the factors in the puzzle before you can figure out what the missing clue should be. That’s about all the mystery I can put in a puzzle like this. Good luck with it!

Since this is different than any other puzzle I’ve ever published, you can see the solution here: # 1366 Fractions Acting Improperly

In elementary school, we learned about improper fractions. Should we call them that? Is it even possible for any kind of number to be IMPROPER? They are simply fractions greater than one. I’ve recently heard the term fraction form used, and ever since I’ve made a point of saying that fractions greater than one are in fraction form. On Twitter, I’ve found a few people who also don’t like using the word improper to describe any fraction.

This first tweet has a link explaining why it is improper to use the term improper fraction:

I hope that you will consider not labeling any fraction as improper, as well!

Now I’ll write a little bit about the number 1366:

• 1366 is a composite number.
• Prime factorization: 1366 = 2 × 683
• 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 1366 has exactly 4 factors.
• Factors of 1366: 1, 2, 683, 1366
• Factor pairs: 1366 = 1 × 1366 or 2 × 683
• 1366 has no square factors that allow its square root to be simplified. √1366 ≈ 36.95944 1366 is also the sum of the twenty-six prime number from 5 to 107. Do you know what all those prime numbers are?

# 1365 Shamrock Mystery

Beautiful shamrocks with their three heart-shaped leaves are not difficult to find. Finding the factors in this shamrock-shaped puzzle might be a different story.  Sure, it might start off to be easy, but after a while, you might find it a wee bit more difficult, unless, of course, the luck of the Irish is with you! Print the puzzles or type the solution in this excel file: 12 Factors 1357-1365

Now I’ll share some information about the number 1365:

• 1365 is a composite number.
• Prime factorization: 1365 = 3 × 5 × 7 × 13
• 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 1365 has exactly 16 factors.
• Factors of 1365: 1, 3, 5, 7, 13, 15, 21, 35, 39, 65, 91, 105, 195, 273, 455, 1365
• Factor pairs: 1365 = 1 × 1365, 3 × 455, 5 × 273, 7 × 195, 13 × 105, 15 × 91, 21 × 65, or 35 × 39
• 1365 has no square factors that allow its square root to be simplified. √1365 ≈ 36.94591 1365 is the hypotenuse of FOUR Pythagorean triples:
336-1323-1365 which is 21 times (16-63-65)
525-1260-1365 which is (5-12-13) times 105
693-1176-1365 which is 21 times (33-56-65)
819-1092-1365 which is (3-4-5) times 273

1365 looks interesting in some other bases:
It’s 10101010101 in BASE 2,
111111 in BASE 4,
2525 in BASE 8, and
555 in BASE 16

I’m feeling pretty lucky that I noticed all those fabulous number facts! If you haven’t been so lucky finding the factors of the puzzle, the same puzzle but with more clues might help: # 1363 and Level 6

The common factors of 60 and 30 allowed in the puzzle are 5, 6, and 10. Which one is the logical choice? Look at the other clues in the puzzle and you should be able to eliminate two of the choices. Print the puzzles or type the solution in this excel file: 12 Factors 1357-1365

Here are a few thoughts about the puzzle number, 1363:

• 1363 is a composite number.
• Prime factorization: 1363 = 29 × 47
• 1363 has no exponents greater than 1 in its prime factorization, so √1363 cannot be simplified.
• The exponents in the prime factorization are 1, and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 1363 has exactly 4 factors.
• The factors of 1363 are outlined with their factor pair partners in the graphic below. 1363 is the hypotenuse of a Pythagorean triple:
940-987-1363 which is (20-21-29) times 47

# What Is Special about √1362

What is special about √1362? I don’t mean to sound like a Chevy commercial, but Just Look At It! I made this gif to show off √1362: make science GIFs like this at MakeaGif
1362 is the smallest number whose square root does that. Yeah, there have been some numbers that came close, but this one includes zero! Thank you Stetson.edu for alerting me to that fact.
Here are a few more facts about the number 1362:
• 1362 is a composite number.
• Prime factorization: 1362 = 2 × 3 × 227
• 1362 has no exponents greater than 1 in its prime factorization, so √1362 cannot be simplified.
• The exponents in the prime factorization are 1, 1, and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 = 8. Therefore 1362 has exactly 8 factors.
• The factors of 1362 are outlined with their factor pair partners in the graphic below.
You can easily calculate a couple of monstrous Pythagorean triples that contain 1362:
1362-463760-463762 calculated from 2(681)(1), 681² – 1²,  681² + 1², and
1362-51520-51538 calculated from 2(227)(3), 227² – 3²,  227² + 3²

# 1361 and Level 5

If you carefully use logic instead of guessing and checking you can find the unique solution to this puzzle without tearing your hair out! Print the puzzles or type the solution in this excel file: 12 Factors 1357-1365

Here are some facts about the number 1361:

• 1361 is a prime number.
• Prime factorization: 1361 is prime.
• 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.

# 1360 and Level 4

There is only one way to write the numbers 1 to 12 in the top row and the first column of this puzzle so that the puzzle could turn into a multiplication table. Can you find the solution? Print the puzzles or type the solution in this excel file: 12 Factors 1357-1365

Here are some facts about the number 1360:

• 1360 is a composite number.
• Prime factorization: 1360 = 2 × 2 × 2 × 2 × 5 × 17, which can be written 1360 = 2⁴ × 5 × 17
• 1360 has at least one exponent greater than 1 in its prime factorization so √1360 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1360 = (√16)(√85) = 4√85
• The exponents in the prime factorization are 4, 1, and 1. Adding one to each exponent and multiplying we get (4 + 1)(1 + 1)(1 + 1) = 5 × 2 × 2 = 20. Therefore 1360 has exactly 20 factors.
• The factors of 1360 are outlined with their factor pair partners in the graphic below. 1360 is the sum of two squares in two different ways:
36² + 8² = 1360
28² + 24² = 1360

1360 is the hypotenuse of FOUR Pythagorean triples:
208-1344-1360 which is 16 times (13-84-85)
576-1232-1360 which is 16 times (36-77-85)
640-1200-1360 which is (8-15-17) times 80
816-1088-1360 which is (3-4-5) times 272

# 1359 and Level 3

The first row with a clue has a 6 in it. Use that 6 and another clue that goes with it, to figure out where to put the factors of 6 in this puzzle. Then work your way down the puzzle, row by row until you have found all the factors of this level 3 puzzle. Print the puzzles or type the solution in this excel file: 12 Factors 1357-1365

Here are a few facts about the number 1359:

• 1359 is a composite number.
• Prime factorization: 1359 = 3 × 3 × 151, which can be written 1359 = 3² × 151
• 1359 has at least one exponent greater than 1 in its prime factorization so √1359 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1359 = (√9)(√151) = 3√151
• The exponents in the prime factorization are 2 and 1. Adding one to each exponent and multiplying we get (2 + 1)(1 + 1) = 3 × 2 = 6. Therefore 1359 has exactly 6 factors.
• The factors of 1359 are outlined with their factor pair partners in the graphic below. 1359 is the difference of two squares in three different ways:
680² – 679² = 1359
228² – 225² = 1359
80² – 71² = 1359

# 1358 and Level 2

If you are familiar with a basic 12 × 12 multiplication table, then you can solve this puzzle. The clues aren’t in the same order as they are in the table, but that only makes it a little more challenging. Print the puzzles or type the solution in this excel file: 12 Factors 1357-1365

Here’s a little bit about the number 1358:

• 1358 is a composite number.
• Prime factorization: 1358 = 2 × 7 × 97
• 1358 has no exponents greater than 1 in its prime factorization, so √1358 cannot be simplified.
• The exponents in the prime factorization are 1, 1, and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 = 8. Therefore 1358 has exactly 8 factors.
• The factors of 1358 are outlined with their factor pair partners in the graphic below. 910-1008-1358 which is 14 times (65-72-97)

Stetson.edu informs us that 1358!!!! + 1 is a prime number.

# 1357 and Level 1

Do you know the multiplication and division facts necessary to solve this puzzle? If you don’t, this puzzle will give you much-needed practice, so give it a try! Print the puzzles or type the solution in this excel file: 12 Factors 1357-1365

Here is some information about the puzzle number, 1357:

• 1357 is a composite number.
• Prime factorization: 1357 = 23 × 59
• 1357 has no exponents greater than 1 in its prime factorization, so √1357 cannot be simplified.
• The exponents in the prime factorization are 1, and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 1357 has exactly 4 factors.
• The factors of 1357 are outlined with their factor pair partners in the graphic below. 1, 3, 5, 7 are the first four odd numbers in order.