1455 and Level 1

Today’s Puzzle:

All of the clues in today’s puzzle are divisible by the same number. Can you figure out what that number is? If you can, then you can solve this puzzle.

Factors of 1455:

That was puzzle number 1455. In case you would like to know a little bit about that number, here are a few facts:

  • 1455 is a composite number.
  • Prime factorization: 1455 = 3 × 5 × 97
  • 1455 has no exponents greater than 1 in its prime factorization, so √1455 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 1455 has exactly 8 factors.
  • The factors of 1455 are outlined with their factor pair partners in the graphic below.

Facts about the number 1455:

1455 is the difference of two squares in four different ways:
728² – 727² = 1455
244² – 241² = 1455
148² – 143² = 1455
56² – 41² = 1455

1455 is also the hypotenuse of four Pythagorean triples:
132-1449-1455 which is 3 times (44-483-485)
279-1428-1455 which is 3 times (93-476-485)
873-1164-1455 which is (3-4-5) times 291
975-1080-1455 which is 15 times (65-72-97)

 

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1443 A Simple Gift for You

This level one puzzle is my simple gift to you. Yes, you can solve it, and you don’t even have to wait until December 25th to discover all the factors and products to be found inside!

That was puzzle number 1443. Here are some facts about that number.

  • 1443 is a composite number.
  • Prime factorization: 1443 = 3 × 13 × 37
  • 1443 has no exponents greater than 1 in its prime factorization, so √1443 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 1443 has exactly 8 factors.
  • The factors of 1443 are outlined with their factor pair partners in the graphic below.

(38-1)(38+1) = 1443 so 1443 is one number away from 38² = 1444.

Actually, 1443 is the difference of two squares in four different ways:
38² – 1² = 1443
62²- 49² = 1443
242² – 239² = 1443
722² – 721² = 1443

1443 is the hypotenuse of FOUR Pythagorean triples:
93-1440-1443 which is 3 times (31-480-481)
468-1365-1443 which is (12-35-37) times 39
555-1332-1443 which is (5-12-13) times 111
957-1080-1443 which is 3 times(319-360-481)

1432 and Level 1

If you can count by tens, then you can solve this Level 1 puzzle. I dare you to prove me wrong!

Print the puzzles or type the solution in this excel file:  10 Factors 1432-1442

Here is some information about the puzzle number, 1432:

  • 1432 is a composite number.
  • Prime factorization: 1432 = 2 × 2 × 2 × 179, which can be written 1432 = 2³ × 179
  • 1432 has at least one exponent greater than 1 in its prime factorization so √1432 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1432 = (√4)(√358) = 2√358
  • The exponents in the prime factorization are 3 and 1. Adding one to each exponent and multiplying we get (3 + 1)(1 + 1) = 4 × 2 = 8. Therefore 1432 has exactly 8 factors.
  • The factors of 1432 are outlined with their factor pair partners in the graphic below.

1432 is 21212 in BASE 5 because
2(5⁴) + 1(5³) + 2(5²) + 1(5¹) + 2(5⁰) = 1432

 

1419 and Level 1

This Level 1 Find the Factors 1-12 puzzle is as simple as clockwork. Can you find all the factors?

Print the puzzles or type the solution in this excel file: 12 Factors 1419-1429

Now I’ll tell you a few things about the puzzle number, 1419:

  • 1419 is a composite number.
  • Prime factorization: 1419 = 3 × 11 × 43
  • 1419 has no exponents greater than 1 in its prime factorization, so √1419 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 1419 has exactly 8 factors.
  • The factors of 1419 are outlined with their factor pair partners in the graphic below.

1419 is the difference of two squares in four different ways:
710² – 709² = 1419
238² – 235² = 1419
70² – 59² = 1419
38² – 5² = 1419

That last one means we are only 25 numbers away from the next perfect square, and that seems like clockwork, too.

1419 is only the second Zeisel number: (105 was the first.)
(4×1-1)(4×3-1)(4×11-1) = 3×11×43 =1419

1410 and Level 1

Start the school year off right with a quick review of the multiplication table. You can actually construct an entire 10 × 10 table with only the nine clues in this puzzle. Figure out where the numbers 1 to 10 go in both the first column and the top row and amaze yourself with how much you remember!

Print the puzzles or type the solution in this excel file: 10 Factors 1410-1418

Here are some facts about the puzzle number, 1410:

  • 1410 is a composite number.
  • Prime factorization: 1410 = 2 × 3 × 5 × 47
  • 1410 has no exponents greater than 1 in its prime factorization, so √1410 cannot be simplified.
  • The exponents in the prime factorization are 1, 1, 1, and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 × 2 = 16. Therefore 1410 has exactly 16 factors.
  • The factors of 1410 are outlined with their factor pair partners in the graphic below.

1410 is the hypotenuse of a Pythagorean triple:
846-1128-1410 which is (3-4-5) times 282.

1391 and Level 1

Many of the clues in this puzzle have double digits. If you know why they do, then you can find all the factors and solve this puzzle!

Print the puzzles or type the solution in this excel file: 12 Factors 1389-1403

Here’s some information about the number 1391:

  • 1391 is a composite number.
  • Prime factorization: 1391 = 13 × 107
  • 1391 has no exponents greater than 1 in its prime factorization, so √1391 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 1391 has exactly 4 factors.
  • The factors of 1391 are outlined with their factor pair partners in the graphic below.

1391 is the hypotenuse of a Pythagorean triple:
535-1284-1391 which is (5-12-13) times 107

1373 and Level 1

This puzzle is really just a multiplication table but the factors are missing and the products are not in order. You can figure out where the factors go, and then the clues will all make sense.


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, 1373:

  • 1373 is a prime number.
  • Prime factorization: 1373 is prime.
  • 1373 has no exponents greater than 1 in its prime factorization, so √1373 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1373 has exactly 2 factors.
  • The factors of 1373 are outlined with their factor pair partners in the graphic below.

How do we know that 1373 is a prime number? If 1373 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1373. Since 1373 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 31, or 37, we know that 1373 is a prime number.

1373 is the sum of two squares:
37² + 2² = 1373

1373 is the hypotenuse of a Pythagorean triple:
148-1365-1373 calculated from 2(37)(2), 37² – 2², 37² + 2²

Here’s another way we know that 1373 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 37² + 2² = 1373 with 37 and 2 having no common prime factors, 1373 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1373. Since 1373 is not divisible by 5, 13, 17,  29, or 37, we know that 1373 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.

1342 Level One Puzzles Only

My son requested I made some pages using only level 1 puzzles for my granddaughter to do. You can see and use the first page above, but I’ve also compiled ALL the level 1 puzzles from 2018 and put them in one excel file: Level 1’s from 2018

Those puzzles will give beginners lots of practice and help them learn the multiplication table.

Now I’ll tell you a little bit about the number 1342:

  • 1342 is a composite number.
  • Prime factorization: 1342 = 2 × 11 × 61
  • 1342 has no exponents greater than 1 in its prime factorization, so √1342 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 1342 has exactly 8 factors.
  • The factors of 1342 are outlined with their factor pairs in the graphic below.

1342 is the hypotenuse of a Pythagorean triple:
242-1320-1342 which is 22 times (11-60-61)

And (11-60-61) times 122 is a triple where 1342 is a leg:
1342-7320-7442. Pretty cool!

The last prime number was 1327 which was 15 numbers ago. The next prime number won’t be until 1361 which is 19 numbers away. Wow! That means that 1342 is 15 numbers away from the CLOSEST prime number, a new record. Thank you Stetson.edu for alerting me to that fact.

1333 and Level 1

How well do you know the multiplication table? If you don’t know it very well, this puzzle can be a fun, no-pressure way to get more familiar with it. Give it a try!

Print the puzzles or type the solution in this excel file: 12 factors 1333-1341

Now I’ll share some facts about the puzzle number, 1333:

  • 1333 is a composite number.
  • Prime factorization: 1333 = 31 × 43
  • 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 1333 has exactly 4 factors.
  • Factors of 1333: 1, 31, 43, 1333
  • Factor pairs: 1333 = 1 × 1333 or 31 × 43
  • 1333 has no square factors that allow its square root to be simplified. √1333 ≈ 36.51027

1333 is the difference of two squares two different ways:
667²-666² = 1333
37² – 6² = 1333