1642 and Level 4

Today’s Puzzle:

There are several clues in this puzzle with more than one factor pair, but if you use logic every step of the way, you can still write the factors 1 to 12 in the appropriate places to turn the puzzle into a multiplication table.

Factors of 1642:

  • 1642 is a composite number.
  • Prime factorization: 1642 = 2 × 821.
  • 1642 has no exponents greater than 1 in its prime factorization, so √1642 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 1642 has exactly 4 factors.
  • The factors of 1642 are outlined with their factor pair partners in the graphic below.

More About the Number 1642:

1642 is the sum of two squares:
39² + 11² = 1642.

1642 is the hypotenuse of a Pythagorean triple:
858-1400-1642, calculated from 2(39)(11), 39² – 11², 39² + 11².
That triple is also 2 times (429-700-821).

1631 The Importance of Practice

Today’s Puzzle:

I did not have the privilege of learning a musical instrument when I was growing up, but I did make sure my children had that opportunity. One of the topics discussed in this next episode of Bill Davidson’s Podcast is the importance that practice plays in both music and mathematics. I thought it was quite good.

I think practicing is best when it is enjoyable. If you solve this musical note puzzle, it will hopefully be an enjoyable way for you to practice a few multiplication and division facts. Just use logic to write the numbers from 1 to 10 in both the first column and the top row so that those numbers and the given clues will function like a multiplication table.

Factors of 1631:

  • 1631 is a composite number.
  • Prime factorization: 1631 = 7 × 233.
  • 1631 has no exponents greater than 1 in its prime factorization, so √1631 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 1631 has exactly 4 factors.
  • The factors of 1631 are outlined with their factor pair partners in the graphic below.

More about the Number 1631:

1631 is the hypotenuse of a Pythagorean triple:
735-1456-1631, which is 7 times (105-208-233).

1631 is the difference of two squares in two different ways:
816² – 815² = 1631, and
120² – 113² = 1631.

I found those number facts just from looking at the factors of 1631.

 

1598 See the Logic in This Level 4 Puzzle

Today’s Puzzle:

Put the numbers from 1 to 10 in both the first column and the top row to turn this level 4 puzzle into a multiplication table. The logic in the ten clues is fairly straight-forward. Go ahead give it a try!

Factors of 1598:

  • 1598 is a composite number.
  • Prime factorization: 1598 = 2 × 17 × 47.
  • 1598 has no exponents greater than 1 in its prime factorization, so √1598 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 1598 has exactly 8 factors.
  • The factors of 1598 are outlined with their factor pair partners in the graphic below.

More about the Number 1598:

1598 is the hypotenuse of a Pythagorean triple:
752-1410-1598, which is (8-15-17) times 94.

1589 Candy Bars

Today’s Puzzle:

Perhaps you can imagine that this level 4 puzzle looks like a couple of candy bars, one for you and one for me!

Factors of 1589:

Divisibility rules let us know quickly that 1589 is not divisible by 2, 3, or 5. Is it divisible by 7, the next prime number. We can apply the divisibility rule for 7 to find out:

1589 is divisible by 7 because
158 – 2(9) = 158 – 18 = 140, and 140 is divisible by 7.

  • 1589 is a composite number.
  • Prime factorization: 1589 = 7 × 227.
  • 1589 has no exponents greater than 1 in its prime factorization, so √1589 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 1589 has exactly 4 factors.
  • The factors of 1589 are outlined with their factor pair partners in the graphic below.

More about the Number 1589:

1589 is the difference of two squares two different ways:
795² – 794² = 1589, and
117² – 110² = 1589.

1589 is the sum of consecutive numbers in three different ways:
1589 is the sum of the two numbers from 794 to 795.
1589 is the sum of the seven numbers from 224 to 230.
1589 is also the sum of the fourteen numbers from 107 to 120,

 

1580 Use Logic to Solve This Puzzle

Today’s Puzzle:

Where should you write the numbers from 1 to 10 in both the first column and the top row to make this puzzle function like a multiplication table? Study it until you find a logical place to start and continue to use logic until you’ve completed the puzzle.

Factors of 1580:

  • 1580 is a composite number.
  • Prime factorization: 1580 = 2 × 2 × 5 × 79, which can be written 1580 = 2² × 5 ×
  • 1580 has at least one exponent greater than 1 in its prime factorization so √1580 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1580 = (√4)(√395) = 2√
  • 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 1580 has exactly 12 factors.
  • The factors of 1580 are outlined with their factor pair partners in the graphic below.

More about the Number 1580:

1580 is the difference of two squares in two different ways:
396² – 394² = 1580, and
84² – 74² = 1580.

1562 Evergreen Tree

Today’s Puzzle:

An evergreen tree doesn’t drop its leaves in the fall or look dead in the winter. As it reminds us of everlasting life, it makes a lovely symbol of Christmas.

Write the numbers from 1 to 10 in both the first column and the top row so that those numbers and the given clues function like a multiplication table. Here is the same puzzle that won’t use as much ink to print:

Factors of 1562:

  • 1562 is a composite number.
  • Prime factorization: 1562 = 2 × 11 × 71.
  • 1562 has no exponents greater than 1 in its prime factorization, so √1562 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 1562 has exactly 8 factors.
  • The factors of 1562 are outlined with their factor pair partners in the graphic below.

Another Fact about the Number 1562:

2(5⁴ + 5³ + 5² + 5¹ + 5⁰) = 1562

1551 Two Straight Lines

Today’s Puzzle:

The clues in this level 4 puzzle form two straight lines, and most of the logic needed to solve it is rather straightforward. Can you find the factors from 1 to 10 without getting twisted up?

Factors of 1551:

1+5 = 6, a multiple of 3, so 1551 is divisible by 3.
Since 1551 is a palindrome with an even number of digits, it is also divisible by 11.

  • 1551 is a composite number.
  • Prime factorization: 1551 = 3 × 11 × 47.
  • 1551 has no exponents greater than 1 in its prime factorization, so √1551 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 1551 has exactly 8 factors.
  • The factors of 1551 are outlined with their factor pair partners in the graphic below.

A Little More about the Number 1551:

Did you notice how many of 1551’s factors are palindromes?

1551 is the difference of two squares in four different ways:
776² – 775² = 1551,
260² – 257² = 1551,
76² – 65² = 1551, and
40² – 7² = 1551. (Yes, we are just 7² or 49 numbers away from 1600, the next perfect square!)

1551 is in this cool pattern:

1529 Pointy Hat

Today’s Puzzle:

A pointy hat is part of many different Halloween costumes: Wizards, Witches, Medieval Princesses, and Clowns come to my mind. Today’s puzzle looks like a pointy hat. Use logic to work your magic in solving it! As always, there is only one solution.

Factors of 1529:

Look at this math fact using the digits of 1529:
1 – 5 + 2 – 9 = -11, a number divisible by 11, so 1529 is divisible by 11.

  • 1529 is a composite number.
  • Prime factorization: 1529 = 11 × 139.
  • 1529 has no exponents greater than 1 in its prime factorization, so √1529 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 1529 has exactly 4 factors.
  • The factors of 1529 are outlined with their factor pair partners in the graphic below.

More about the Number 1529:

1529 is the difference of two squares in two different ways:
765² – 764²  = 1529, and
75² – 64²  = 1529.

 

1516 and Level 4

Today’s Puzzle:

Using logic, write each number from 1 to 12 in both the first column and the top row so that those numbers are the factors of the given clues.

Factors of 1516:

  • 1516 is a composite number.
  • Prime factorization: 1516 = 2 × 2 × 379, which can be written 1516 = 2² × 379
  • 1516 has at least one exponent greater than 1 in its prime factorization so √1516 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1516 = (√4)(√379) = 2√379
  • 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 1516 has exactly 6 factors.
  • The factors of 1516 are outlined with their factor pair partners in the graphic below.

One More Fact about the Number 1516:

Only one of its factor pairs add up to an even number, so 1516 is the difference of two squares in only one way:
380² – 378² = 1516.

1493 and Level 4

Today’s Puzzle:

Write the numbers from 1 to 12 in both the first column and the top row so that those numbers are the factors of the clues given in the puzzle:

Factors of 1493:

  • 1493 is a prime number.
  • Prime factorization: 1493 is prime.
  • 1493 has no exponents greater than 1 in its prime factorization, so √1493 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1493 has exactly 2 factors.
  • The factors of 1493 are outlined with their factor pair partners in the graphic below.

How do we know that 1493 is a prime number? If 1493 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1493. Since 1493 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, or 37, we know that 1493 is a prime number.

Other Facts about the number 1493:

1493 is the last prime number in the fourth prime quintuplet,
(1481, 1483, 1487, 1489, 1493), which is the smallest prime quintuplet that is not also part of a prime sextuplet.
In prime quintuplets, the first three numbers, the middle three numbers, and the last three numbers each form a prime triplet. Thus,1493 is the last prime number in the third prime triplet formed from the numbers in the fourth prime quintuplet.

1493 is the sum of two squares:
38² + 7² = 1493

1493 is the hypotenuse of a Pythagorean triple:
532-1395-1493, calculated from 2(38)(7), 38² – 7², 38² + 7²

Here’s another way we know that 1493 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 38² + 7² = 1493 with 38 and 7 having no common prime factors, 1493 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1493. Since 1493 is not divisible by 5, 13, 17, 29, or 37, we know that 1493 is a prime number.