# 1401 Roasting Over an Open Fire

I went camping last week. My family roasted hotdogs. Some people refer to them as mystery meat. Others roasted marshmallows. I was surprised to learn that almost all brands of marshmallows have blue dye in them.  I’m told that without that blue dye the marshmallows will lose their whiteness as they sit on store shelves. Why they have to be that white is a mystery to me.

Here’s a mystery level puzzle for you to solve. It looks a lot like the utensil that was used to roast the hotdogs and marshmallows. Print the puzzles or type the solution in this excel file: 12 Factors 1389-1403

Now I’ll tell you something about the number 1401:

• 1401 is a composite number.
• Prime factorization: 1401 = 3 × 467
• 1401 has no exponents greater than 1 in its prime factorization, so √1401 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 1401 has exactly 4 factors.
• The factors of 1401 are outlined with their factor pair partners in the graphic below. 1401 is the difference of two squares in two different ways. Can you figure out what those ways are?

# 1388 Mystery Level

Sometimes puzzles start out easy enough but get a little more complicated later on. Does that happen with this puzzle? There’s only one way to find out! Print the puzzles or type the solution in this excel file: 10 Factors 1373-1388

Here are some facts about the number 1388:

• 1388 is a composite number.
• Prime factorization: 1388 = 2 × 2 × 347, which can be written 1388 = 2² × 347
• 1388 has at least one exponent greater than 1 in its prime factorization so √1388 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1388 = (√4)(√347) = 2√347
• 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 1388 has exactly 6 factors.
• The factors of 1388 are outlined with their factor pair partners in the graphic below. 1388 is the difference of two squares:
348² – 346² = 1388

# 1385 Mystery Level

You can suspect that the common factor of 9 and 6 is either 1 or 3, but don’t jump to conclusions about which one will satisfy this mystery! There’s important evidence elsewhere in the puzzle that you should consider first. Print the puzzles or type the solution in this excel file: 10 Factors 1373-1388

Now I’ll share some facts about the puzzle number, 1385:

• 1385 is a composite number.
• Prime factorization: 1385 = 5 × 277
• 1385 has no exponents greater than 1 in its prime factorization, so √1385 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 1385 has exactly 4 factors.
• The factors of 1385 are outlined with their factor pair partners in the graphic below. 1385 is the sum of two squares in two different ways:
32² + 19² = 1385
37² + 4² = 1385

1385 is the hypotenuse of a Pythagorean triple:
296-1353-1385 calculated from 2(37)(4), 37² – 4², 37² + 4²
575-1260-1385 which is 5 times (115-252-277)
663-1216-1385 calculated from 32² – 19², 2(32)(19), 32² + 19²
831-1108-1385 which is (3-4-5) times 277

# 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: # 1356 Mystery

The clues that appear in this puzzle are all you need to find all the factors. Seriously. Some mysteries are easier to solve than others. Give this one a try! Print the puzzles or type the solution in this excel file: 10 Factors 1347-1356

Here is some information about the puzzle number, 1356:

• 1356 is a composite number.
• Prime factorization: 1356 = 2 × 2 × 3 × 113, which can be written 1356 = 2² × 3 × 113
• 1356 has at least one exponent greater than 1 in its prime factorization so √1356 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1356 = (√4)(√339) = 2√339
• The exponents in the prime factorization are 2, 1, and 1. Adding one to each exponent and multiplying we get (2 + 1)(1 + 1)(1 + 1) = 3 × 2 × 2 = 12. Therefore 1356 has exactly 12 factors.
• The factors of 1356 are outlined with their factor pair partners in the graphic below. 1356 is the hypotenuse of a Pythagorean triple:
180-1344-1356 which is 12 times (15-112-113)

# 1341 Not a Pretty Picture

In real mysteries, sometimes clues are close together, sometimes they are far apart. They don’t often make a pretty picture. Can you use the clues and logic to solve this mystery? Print the puzzles or type the solution in this excel file: 12 factors 1333-1341

Here is some information about the number 1341:

• 1341 is a composite number.
• Prime factorization: 1341 = 3 × 3 × 149, which can be written 1341 = 3² × 149
• 1341 has at least one exponent greater than 1 in its prime factorization so √1341 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1341 = (√9)(√149) = 3√149
• 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 1341 has exactly 6 factors.
• The factors of 1341 are outlined with their factor pairs in the graphic below. 1341 is the sum of two squares:
30² + 21² = 1341

1341 is the hypotenuse of a Pythagorean triple:
459-1260-1341 which is 9 times (51-140-149) and
can also be calculated from 30² – 21², 2(30)(21), 30² + 21²

# 1340 Mystery Level

Which two of these clues will use both of the 1’s? 10, 12, 9, or 7. I’ll never tell. You will have to solve that mystery yourself! Print the puzzles or type the solution in this excel file: 12 factors 1333-1341

Here are a few facts about the number 1340:

• 1340 is a composite number.
• Prime factorization: 1340 = 2 × 2 × 5 × 67, which can be written 1340 = 2² × 5 × 67
• 1340 has at least one exponent greater than 1 in its prime factorization so √1340 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1340 = (√4)(√335) = 2√335
• The exponents in the prime factorization are 2, 1, and 1. Adding one to each exponent and multiplying we get (2 + 1)(1 + 1)(1 + 1) = 3 × 2 × 2 = 12. Therefore 1340 has exactly 12 factors.
• The factors of 1340 are outlined with their factor pairs in the graphic below. 1340 is the hypotenuse of a Pythagorean triple:
804-1072-1340 which is (3-4-5) times 268

# 1332 Yet Another Christmas Tree

Here is yet another Christmas tree for you to enjoy this holiday season.

Print the puzzles or type the solution in this excel file:10-factors-1321-1332

Here are a few facts about the number 1332:

• 1332 is a composite number.
• Prime factorization: 1332 = 2 × 2 × 3 × 3 × 37, which can be written 1332 = 2² × 3² × 37
• The exponents in the prime factorization are 2, 2 and 1. Adding one to each and multiplying we get (2 + 1)(2 + 1)(1 + 1) = 3 × 3 × 2 = 18. Therefore 1332 has exactly 18 factors.
• Factors of 1332: 1, 2, 3, 4, 6, 9, 12, 18, 36, 37, 74, 111, 148, 222, 333, 444, 666, 1332
• Factor pairs: 1332 = 1 × 1332, 2 × 666, 3 × 444, 4 × 333, 6 × 222, 9 × 148, 12 × 111, 18 × 74 or 36 × 37
• Taking the factor pair with the largest square number factor, we get √1332 = (√36)(√37) = 6√37 ≈ 36.49658

Here are a couple of factor trees for 1332: Since 36 × 37 = 1332, we know that 1332 is the sum of the first 36 even numbers. (The first 36 numbers add up to the infamous 666, and 2 times 666 is 1332.)

Because 1332 is divisible by both 3 and 37, it has several repdigits as factors, 111, 222, 333, 444, and 666.

1332 is the sum of four consecutive prime numbers:
317 + 331 + 337 + 347 = 1332

1332 is the sum of two square numbers:
36² + 6² = 1332

1332 is the hypotenuse of a Pythagorean triple:
432-1260-1332 which is 36 times (12-35-37)
It can also be calculated from 2(36)(6), 36² – 6², 36² + 6²

The Gingerbread man can be tricky so be careful while solving this puzzle. He has fooled and run away from many different people and animals. The mystery is can YOU outfox this one?

Print the puzzles or type the solution in this excel file:10-factors-1321-1332

The factoring information for the number 1324 is far less tricky than you might imagine:

• 1324 is a composite number.
• Prime factorization: 1324 = 2 × 2 × 331, which can be written 1324 = 2² × 331
• 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 1324 has exactly 6 factors.
• Factors of 1324: 1, 2, 4, 331, 662, 1324
• Factor pairs: 1324 = 1 × 1324, 2 × 662, or 4 × 331
• Taking the factor pair with the largest square number factor, we get √1324 = (√4)(√331) = 2√331 ≈ 36.38681

1324 is palindrome 1M1 in BASE 27 (M is 22 base 10) because
27² + 22(27) + 1 = 1324

# 1318 Another Candy Cane

Candy canes are rarely alone. They almost always have a twin close-by. Nevertheless, this mystery-level candy cane puzzle only looks similar to the previous one. You will have to consider completely different factors to solve it.

Print the puzzles or type the solution in this excel file: 12 factors 1311-1319

Here are a few facts about the number 1318:

• 1318 is a composite number.
• Prime factorization: 1318 = 2 × 659
• 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 1318 has exactly 4 factors.
• Factors of 1318: 1, 2, 659, 1318
• Factor pairs: 1318 = 1 × 1318 or 2 × 659
• 1318 has no square factors that allow its square root to be simplified. √1318 ≈ 36.30427

1318 is palindrome 414 in BASE 18