1482 and Level 4

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Today’s Puzzle:

Can you write all the numbers from 1 to 10 in both the first column and the top row so that those numbers become the factors of the given clues?

Factors of 1482:

  • 1482 is a composite number.
  • Prime factorization: 1482 = 2 × 3 × 13 × 19
  • 1482 has no exponents greater than 1 in its prime factorization, so √1482 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 1482 has exactly 16 factors.
  • The factors of 1482 are outlined with their factor pair partners in the graphic below.

Another Fact about the Number 1482:

1482 is the hypotenuse of a Pythagorean triple:
570-1368-1482 which is (5-12-13) times 114.

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1481 is the First Prime Number in the Fifth Prime Decade

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What Kind of Prime Number is 1481?

1481 is the 234th prime number. It is part of a twin prime pair, a prime triplet, a prime quadruplet, and even a prime quintuplet. All of those designations are nice, but to me, the most beautiful is the prime decade (the prime quadruplets not counting (5, 7, 11, 13)).

1481 is the first prime number in the fifth prime decade. The last digits of the numbers in every prime decade make a lovely pattern: 1, 3, 7, and 9. Once you know the first prime number in the decade, you also know the other three! That’s beautiful!

I found the numbers in the first five prime decades easy to memorize. We start with the teen decade followed by the first and last decades in the 100’s. Then199 is almost 200 and that helps me remember 821, and 821 helps me remember 1481.

Factors of 1481:

  • 1481 is a prime number.
  • Prime factorization: 1481 is prime.
  • 1481 has no exponents greater than 1 in its prime factorization, so √1481 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1481 has exactly 2 factors.
  • The factors of 1481 are outlined with their factor pair partners in the graphic below.

How do we know that 1481 is a prime number? If 1481 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1481. Since 1481 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 31, or 37, we know that 1481 is a prime number.

More about the Number 1481:

1481 is the sum of two squares:
35² + 16² = 1481.

1481 is the hypotenuse of a Pythagorean triple:
969-1120-1481, calculated from 35² – 16², 2(35)(16), 35² + 16².

Here’s another way we know that 1481 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 35² + 16² = 1481 with 35 and 16 having no common prime factors, 1481 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1481. Since 1481 is not divisible by 5, 13, 17, 29, or 37, we know that 1481 is a prime number.

 

1480 and Level 3

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Today’s Puzzle:

This is a level 3 puzzle. Start with the common factor of 21 and 14, then fill in the cells in the first column in order from top to bottom as well as the factors in the top row until you have placed all of the factors from 1 to 10 in both of those places.

Factors of 1480:

80 is divisible by 8 and 4 is even, so 1480 is divisible by 8. Here’s a little more about its factors:

  • 1480 is a composite number.
  • Prime factorization: 1480 = 2 × 2 × 2 × 5 × 37, which can be written 1480 = 2³ × 5 × 37
  • 1480 has at least one exponent greater than 1 in its prime factorization so √1480 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1480 = (√4)(√370) = 2√370
  • The exponents in the prime factorization are 3,1 and 1. Adding one to each exponent and multiplying we get (3 + 1)(1 + 1) (1 + 1) = 4 × 2 × 2 = 16. Therefore 1480 has exactly 16 factors.
  • The factors of 1480 are outlined with their factor pair partners in the graphic below.

Something else about the number 1480:

1480 is the hypotenuse of FOUR Pythagorean triples:
456-1408-1480 which is 8 times (57-176-185)
480-1400-1480 which is (12-35-37) times 40
832-1224-1480 which is 8 times (104-153-185)
888-1184-1480 which is (3-4-5) times 296

1479 and Level 2

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Today’s Puzzle:

Can you find the factors from 1 to 10 that will work with the clues in this puzzle to make a multiplication table?

Factors of 1479:

  • 1479 is a composite number.
  • Prime factorization: 1479 = 3 × 17 × 29
  • 1479 has no exponents greater than 1 in its prime factorization, so √1479 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 1479 has exactly 8 factors.
  • The factors of 1479 are outlined with their factor pair partners in the graphic below.

Another Fact about the Number 1479:

1479 is the hypotenuse of FOUR Pythagorean triples:
396-1425-1479 which is 3 times (132-475-493)
465-1404-1479 which is 3 times (155-468-493)
696-1305-1479 which is (8-15-17) times 87
1020-1071-1479 which is (20-21-29) times 51

1478 and Level 1

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Today’s Puzzle:

If you can divide a few even numbers in half, then you can find all the factors needed to solve this puzzle. Then you can complete the multiplication table that the puzzle makes. Go for it!

Factors of 1478:

  • 1478 is a composite number.
  • Prime factorization: 1478 = 2 × 739
  • 1478 has no exponents greater than 1 in its prime factorization, so √1478 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 1478 has exactly 4 factors.
  • The factors of 1478 are outlined with their factor pair partners in the graphic below.

A little more about the number 1478:

1478 is not the sum or the difference of two squares, but it is the short leg in a Pythagorean triple:
(1478-546120-546122) calculated from 2(739)(1), 739² – 1², 739² + 1².

1477 Challenge

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Today’s Puzzle:

It’s been a few days since I’ve last looked at this puzzle, but I was still able to solve it in 17 minutes. Can you beat my time?

Print the puzzles or type the solution in this excel file: 12 Factors 1468-1477

Factors of 1477:

  • 1477 is a composite number.
  • Prime factorization: 1477 = 7 × 211
  • 1477 has no exponents greater than 1 in its prime factorization, so √1477 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 1477 has exactly 4 factors.
  • The factors of 1477 are outlined with their factor pair partners in the graphic below.

Another Fact about the Number 1477:

1477 is the difference of two squares in two different ways:
739² – 738² = 1477
109² – 102² = 1477

1476 Mystery

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Today’s Puzzle:

To solve this mystery puzzle, first, gather some facts. I mean, common factors:
Common Factors of 6, 3, 12, and 9 are 1 and 3.
For 20 and 10 we have 2, 5, and 10.
And for 20 and 60, we can only use 5 or 10 because common factor, 20, is too big for a 1 to 12 puzzle.

Which common factors “Done it”? Don’t jump to conclusions. Remember, a good detective will use the facts and logic to figure out the mystery. Good luck!

Factors of 1476:

  • 1476 is a composite number.
  • Prime factorization: 1476 = 2 × 2 × 3 × 3 × 41, which can be written 1476 = 2² × 3² × 41
  • 1476 has at least one exponent greater than 1 in its prime factorization so √1476 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1476 = (√36)(√41) = 6√41
  • The exponents in the prime factorization are 2, 2, and 1. Adding one to each exponent and multiplying we get (2 + 1)(2 + 1)(1 + 1) = 3 × 3 × 2 = 18. Therefore 1476 has exactly 18 factors.
  • The factors of 1476 are outlined with their factor pair partners in the graphic below.

Simplifying √1476 doesn’t have to be a mystery! Here’s one strategy to do it:

More Facts about the number 1476:

1476 is the sum of two squares:
30² + 24² = 1476

1476 is the hypotenuse of a Pythagorean triple:
324-1440-1476 which is 36 times (9-40-41),
and can also be calculated from 30² – 24², 2(30)(24), 30² + 24².

 

1475 The Logic Needed to Solve This Mystery

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Today’s Puzzle:

Solving this puzzle will be as easy as 1-2-3 if you begin by asking yourself which clues must use the 1’s, the 2’s, and the 3’s, but not necessarily in that order. At the same time, you can eliminate two common factors of 48 and 72, and finishing the puzzle should then become fairly routine.

Factors of 1475:

  • 1475 is a composite number.
  • Prime factorization: 1475 = 5 × 5 × 59, which can be written 1475 = 5² × 59
  • 1475 has at least one exponent greater than 1 in its prime factorization so √1475 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1475 = (√25)(√59) = 5√59
  • 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 1475 has exactly 6 factors.
  • The factors of 1475 are outlined with their factor pair partners in the graphic below.

Another Fact about the Number 1475:

1475 is the hypotenuse of two Pythagorean triples:
413-1416-1475 which is (7-24-25) times 59.
885-1180-1475 which is (3-4-5) times 295.

 

1474 and Level 6

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Today’s Puzzle:

If you think that only kids will find the Find the Factors puzzles a challenge, then this puzzle will likely change your mind. Seriously, can you use logic to find where the factors from 1 to 12 must go to complete this puzzle? There is a reason this is a level 6 puzzle!

Factors of 1474:

The last digit of 1474 is even so 1474 is divisible by 2. Yes, that was easy, but try this next one:
1 – 4 + 7 – 4 = 0, so 1474 can be evenly divided by 11.

  • 1474 is a composite number.
  • Prime factorization: 1474 = 2 × 11 × 67
  • 1474 has no exponents greater than 1 in its prime factorization, so √1474 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 1474 has exactly 8 factors.
  • The factors of 1474 are outlined with their factor pair partners in the graphic below.

Another Fact about the Number 1474:

1474 can not be written as the sum of two squares or the difference of two squares, but it is still the short leg in two Pythagorean triples. Here is one of them:
1474-4368-4610, calculated from 2(67)(11), 67² – 11², 67² + 11²

 

1473 A Level 5 Designed by My Granddaughter

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Today’s Puzzle:

My son and his family arrived today for a two-month long visit. We are so excited that they are here. His nine-year-old daughter has solved her share of Find the Factors puzzles. Today I gave her the opportunity to help me design one which she thoroughly loved doing. Here is the first of her creations. Will you be able to solve it? It’s a level 5 so you might find it a bit tricky.

Factors of 1473:

  • 1473 is a composite number.
  • Prime factorization: 1473 = 3 × 491
  • 1473 has no exponents greater than 1 in its prime factorization, so √1473 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 1473 has exactly 4 factors.
  • The factors of 1473 are outlined with their factor pair partners in the graphic below.

More about the number 1473:

1473 can be written as the difference of two squares in two different ways:
737² – 736² = 1473
247² – 244² = 1473