# 1472 and Level 4

### Today’s Puzzle:

Can you use logic to figure out where the factors from 1 to 12 must go to make the given clues be the products of those factors? ### Factors of 1472:

• 1472 is a composite number.
• Prime factorization: 1472 = 2 × 2 × 2 × 2 × 2 × 2 × 23, which can be written 1472 = 2⁶ × 23.
• 1472 has at least one exponent greater than 1 in its prime factorization so √1472 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1472 = (√64)(√23) = 8√23
• The exponents in the prime factorization are 6 and 1. Adding one to each exponent and multiplying we get (6 + 1)(1 + 1) = 7 × 2 = 14. Therefore 1472 has exactly 14 factors.
• The factors of 1472 are outlined with their factor pair partners in the graphic below. ### A Little More about the Number 1472:

1472 is the difference of two squares in FIVE different ways:
369² – 367² = 1472
186² – 182² = 1472
96² – 88² = 1472
54² – 38² = 1472
39² – 7² = 1472

# 1459 and Level 4

### Today’s Puzzle:

Can you place the numbers 1 to 10 in both the first column and the top row so that those numbers are the factors of the given clues? You might surprise yourself with how well you do! ### Factors of 1459:

• 1459 is a prime number.
• Prime factorization: 1459 is prime.
• 1459 has no exponents greater than 1 in its prime factorization, so √1459 cannot be simplified.
• The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1459 has exactly 2 factors.
• The factors of 1459 are outlined with their factor pair partners in the graphic below. How do we know that 1459 is a prime number? If 1459 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1459. Since 1459 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 31, or 37, we know that 1459 is a prime number.

### Other Facts about the number 1459:

1459 is the difference of two squares:
730² – 729² = 1459

From Stetson.edu we learn this chilling cubic fact:
1³ + 4³ + 5³ + 9³ = 919, and
9³ + 1³ + 9³ = 1459

# 1446 Peppermint Stick

Red and green striped peppermint sticks are often seen in stores and homes in December. Can you lick this peppermint stick puzzle or will you let it lick you? The puzzle number was 1446. Here are a few facts about that number:

• 1446 is a composite number.
• Prime factorization: 1446 = 2 × 3 × 241
• 1446 has no exponents greater than 1 in its prime factorization, so √1446 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 1446 has exactly 8 factors.
• The factors of 1446 are outlined with their factor pair partners in the graphic below. 1446 is also the hypotenuse of a Pythagorean triple:
720-1254-1446 which is 6 times (120-209-241)

# 1435 is the 23rd Friedman Number

41 ×35 = 1435. Since the same digits are used on both sides of the equal sign that makes 1435 the 23rd Friedman number.

Since it’s good for puzzles to have a puzzle number to distinguish them from each other, I’ve given this one the number 1435. Can you solve it? Print the puzzles or type the solution in this excel file:  10 Factors 1432-1442

Here are a few more facts about the number 1435:

• 1435 is a composite number.
• Prime factorization: 1435 = 5 × 7 × 41
• 1435 has no exponents greater than 1 in its prime factorization, so √1435 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 1435 has exactly 8 factors.
• The factors of 1435 are outlined with their factor pair partners in the graphic below. 1435 is also the hypotenuse of FOUR Pythagorean triples:
315-1400-1435 which is 35 times (9-40-41)
588-1309-1435 which is 7 times (84-187-205)
861-1148-1435 which is (3-4-5) times 287
931-1092-1435 which is 7 times (133-156-205)

# 1413 and Level 4

You find a rhythm as you solve this level 4 puzzle. The logic is quite easy for most of it, but there is at least one place that will require you to think things through before proceeding. Print the puzzles or type the solution in this excel file: 10 Factors 1410-1418

Now I’ll share a few facts about the puzzle number, 1413:

• 1413 is a composite number.
• Prime factorization: 1413 = 3 × 3 × 157, which can be written 1413 = 3² × 157
• 1413 has at least one exponent greater than 1 in its prime factorization so √1413 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1413 = (√9)(√157) = 3√157
• 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 1413 has exactly 6 factors.
• The factors of 1413 are outlined with their factor pair partners in the graphic below. 1413 is the sum of two squares:
33² + 18² = 1413

That means that 1413 is the hypotenuse of a Pythagorean triple:
765-1188-1413 calculated from 33² – 18², 2(33)(18), 33² + 18².
It is also 9 times (85-132-157)

# 1397 and Level 4

I bet you know enough multiplication facts to get this puzzle started. Once you’ve started it, you might as well finish it. You will feel so clever when you do! Print the puzzles or type the solution in this excel file: 12 Factors 1389-1403

Now I’ll write a little bit about the puzzle number, 1397:

• 1397 is a composite number.
• Prime factorization: 1397 = 11 × 127
• 1397 has no exponents greater than 1 in its prime factorization, so √1397 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 1397 has exactly 4 factors.
• The factors of 1397 are outlined with their factor pair partners in the graphic below. 1397 is the difference of two squares two different ways:
699² – 698² = 1397
69² – 58² = 1397

# 1382 and Level 4

After solving a couple of level 3 puzzles, you are ready to give a level 4 puzzle a try. It isn’t any more difficult to solve than a level 3, except that the clues are not given in a logical order. Don’t let that stop you from succeeding! Print the puzzles or type the solution in this excel file: 10 Factors 1373-1388

Now I’ll share some information about the number 1382:

• 1382 is a composite number.
• Prime factorization: 1382 = 2 × 691
• 1382 has no exponents greater than 1 in its prime factorization, so √1382 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 1382 has exactly 4 factors.
• The factors of 1382 are outlined with their factor pair partners in the graphic below. 1382 is in one Pythagorean triple:
1382-477480-477482 calculated from 2(691)(1), 691² – 1², 691² + 1²

# 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

# 1353 How to Solve a Level 4 Puzzle

Try solving this level 4 puzzle. If you need help with it, I explain the steps in the video below the puzzle: Print the puzzles or type the solution in this excel file: 10 Factors 1347-1356

Now I’ll tell you a little bit about the puzzle number, 1353:

• 1353 is a composite number.
• Prime factorization: 1353 = 3 × 11 × 41
• 1353 has no exponents greater than 1 in its prime factorization, so √1353 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 1353 has exactly 8 factors.
• The factors of 1353 are outlined with their factor pair partners in the graphic below. 1353 is the hypotenuse of a Pythagorean triple:
297-1320-1353 which is 33 times (9-40-41)

# 1337 and Level 4

If you can solve a level 3 puzzle, give this one a try. You will have to locate the best clues to use in the beginning and then what is the most logical clue to use next over and over, but I think you can do that! Print the puzzles or type the solution in this excel file: 12 factors 1333-1341

Here is some information about the puzzle number, 1337:

• 1337 is a composite number.
• Prime factorization: 1337 = 7 × 191
• 1337 has no exponents greater than 1 in its prime factorization, so √1337 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 1337 has exactly 4 factors.
• The factors of 1337 are outlined with their factor pairs in the graphic below. 1337 is the difference of two squares two ways:
99² – 92² = 1337
669² – 668² = 1337