Why Do Factor Pairs of 1560 Make Sum-Difference?

Today’s Puzzle:

1560 has 16 different factor pairs. One of those pairs sum up to 89, and another pair subtracts to 89. It is only the 50th time that the sum of a factor pair of a number equals the difference of one of its other factor pairs.

You may have seen other Sum-Difference Puzzles where I’ve paired one puzzle with another puzzle and mentioned that the second puzzle was really the first puzzle in disguise. That is not the case for this puzzle because 89 is a prime number. This puzzle is a primitive; there is not a simpler puzzle that is its equivalent. Don’t let that worry you, however, everything you need to solve this puzzle can be found in this post.

Although I am making a big deal about our number 1560, it is the 89 which allows this puzzle to exist in the first place. You see, 89 is the hypotenuse of a Pythagorean triple, (39-80-89), and thus, 39² + 80² = 89². Since that triple is a primitive, the sum-difference is a primitive as well.

Note that (40)(39) = 1560.
We want to use the quadratic formula to solve 40x² + 89x + 39 = 0 and 40x² + 89x – 39 = 0. Let’s combine the left sides of those two equations into one expression:
40x² + 89x ± 39.
The discriminant would be 89² – 4(40)(±39)
= 89² ± 4(40)(39)
= 89² ± 2(80)(39)
= 39² + 80² ± 2(80)(39), ( That’s because 39² + 80² = 89².)
= 80² ± 2(80)(39) + 39²
= (80 ± 39)², a perfect square!

That perfect square makes 1560 one of those relatively rare numbers with factor pairs that make sum-difference.

Factors of 1560:

  • 1560 is a composite number.
  • Prime factorization: 1560 = 2 × 2 × 2 × 3 × 5 × 7, which can be written 1560 = 2³ × 3 × 5 × 7.
  • 1560 has at least one exponent greater than 1 in its prime factorization so √1560 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1560 = (√4)(√390) = 2√390
  • The exponents in the prime factorization are 2, 1, and 2. Adding one to each exponent and multiplying we get (3 + 1)(1 + 1)(1 + 1) (1 + 1) = 4 × 2 × 2 × 2 = 32. Therefore 1560 has exactly 32 factors.
  • The factors of 1560 are outlined with their factor pair partners in the graphic below.

That’s a lot of factor pairs for one number! Here’s a graphic showing those same factor pairs but with their sums and differences included:

Find the 89 in the Sum column and the 89 in the Difference column, and you will see the factor pairs by those 89’s that make the sum and difference needed to solve the puzzle.

More about the Number 1560:

1560 = 39 × 40, so 1560 is the sum of the first 39 even numbers:
2 + 4 + 6 + 8 + . . . + 74 + 76 + 78 = 1560.

1560 is the hypotenuse of FOUR Pythagorean triples:
384-1512-1560, which is 24 times (16-63-65),
600-1440-1560, which is (5-12-13) times 120,
792-1344-1560, which is 24 times (33-56-65), and
936-1248-1560, which is (3-4-5) times 312.

Since 1560 is a hypotenuse four different ways, could it be the bottom part of four different Sum-Difference puzzles? Yes!

You can find the top part of the puzzle by finding one half of the product of the first two numbers in each triple:
384 × 1512 ÷ 2 = ____________,
600 × 1440 ÷ 2 = ____________,
792 × 1344 ÷ 2 = ____________,
936 × 1248 ÷ 2 = ____________.

The numbers that go in the blanks are all between 100 thousand and one million, but each one of those numbers will have a factor pair that adds up to 1560 as well as another one that subtracts to 1560. If you find them, go ahead and brag about it! It will be quite an accomplishment!

A Factor Tree for 1560:

Here is one of MANY possible factor trees for 1560:

 

Factors of 1536 Make Sum-Difference!

Today’s Puzzles:

1536 has 10 different factor pairs. One of those pairs adds up to 80 and a different one subtracts to give 80. Can you find the factor pairs that make sum-difference and write them in the puzzle? You can look at all of the factor pairs of 1536 in the graphic after the puzzle, but the second puzzle is really just the first puzzle in disguise. So try solving that easier puzzle first.

That first puzzle is the first possible Sum-Difference puzzle. The second puzzle is only the 49th possible puzzle.

Factors of 1536:

  • 1536 is a composite number.
  • Prime factorization: 1536 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 3, which can be written 1536 = 2⁹ × 3.
  • 1536 has at least one exponent greater than 1 in its prime factorization so √1536 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1536 = (√256)(√6) = 16√6
  • The exponents in the prime factorization are 9 and 1. Adding one to each exponent and multiplying we get (9 + 1)(1 + 1) = 10 × 2 = 20. Therefore 1536 has exactly 20 factors.
  • The factors of 1536 are outlined with their factor pair partners in the graphic below.

More About the Number 1536:

1536 is the difference of two squares in EIGHT different ways!
385² – 383² = 1536,
194² – 190² = 1536,
131² – 125² = 1536,
100² – 92² = 1536,
70² – 58² = 1536,
56² – 40² = 1536,
44² – 20² = 1536, and
40² – 8² = 1536.

1536 = 6 × 256.
256 is the 100th number whose square root can be simplified.
1536 is the 600th number whose square root can be simplified.

Here are the 501st to the 600th simplifiable square roots. If at least three square roots are consecutive, they are highlighted.

 

Celebrating 1500 with a Horse Race and Much More!

Pick Your Pony!

Writing 1500 posts is quite a milestone. I’ll begin the celebration with an exciting horse race! Let me explain:

Each prime number has exactly 2 factors. Every composite number between 1401 and 1500 has somewhere between 4 and 36 factors. Which quantity of factors do you think will appear most often for these numbers? Pick that amount as your pony and see how far it gets in this horse race!
1401 to 1500 Horse Race

make science GIFs like this at MakeaGif

Did the race results surprise you? They surprised me!

Prime Factorization for numbers from 1401 to 1500:

Here’s a chart showing the prime factorization of all those numbers and the amount of factors each number has. Numbers in pink have exponents in their prime factorization so their square roots can be simplified:

Today’s Puzzles:

Let’s continue the celebration with a puzzle: 1500 has 12 different factor pairs. One of those pairs adds up to 85 and one of them subtracts to give 85. Can you find those factor pairs that make sum-difference and write them in the puzzle? You can look at all of the factor pairs of 1500 in the graphic after the puzzle, but the second puzzle is really just the first puzzle in disguise. So try solving that easier puzzle first.

Factors of 1500:

  • 1500 is a composite number.
  • Prime factorization: 1500 = 2 × 2 × 3 × 5 × 5 × 5, which can be written 1500 = 2² × 3 × 5³
  • 1500 has at least one exponent greater than 1 in its prime factorization so √1500 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1500 = (√100)(√15) = 10√15
  • The exponents in the prime factorization are 2, 1, and 3. Adding one to each exponent and multiplying we get (2 + 1)(1 + 1)(3 + 1) = 3 × 2 × 4 = 24. Therefore 1500 has exactly 24 factors.
  • The factors of 1500 are outlined with their factor pair partners in the graphic below.

A Forest of Factor Trees:

Take a few minutes to hike in this forest featuring a few of the MANY possible factor trees for 1500. Celebrate each tree’s uniqueness!

Other Facts to Celebrate about 1500:

Oeis.org tells us that  (5+1) × (5+5) × (5+0) × (5+0) = 1500.

1500 is the hypotenuse of THREE Pythagorean triples:
420-1440-1500, which is (7-24-25) times 60,
528-1404-1500, which is 12 times (44-117-225),
900-1200-1500, which is (3-4-5) times 300.

1470 Can You Find Factor Pairs That Make Sum-Difference?

Today’s Puzzles:

I bet you can find a factor pair of 30 that adds up to 13 as well as another factor pair of 30 that subtracts to give you 13.

If you can solve that simple puzzle, then you will be able to solve the puzzle next to it. Even though 1470 has 12 different factor pairs, you don’t have to worry too much about them: All of the answers to the second puzzle are just _____ times the answers to the first puzzle! (And 1470 is _____² times 30.)

Likewise, don’t get scared off with this next set of puzzles one of which wants you to find the factor pairs of 518616 that add up or subtract to 1470. Crazy, right? Again, if you can solve the first puzzle in the set, and if you can multiply a 3-digit number by a 1-digit number, you can easily solve the second puzzle because the answers are just _________ times the answers to the first puzzle in the set! (And 518616 is merely _________² times 6.)

If you need more help than what I’ve already said, scroll down to the factor trees for 1470. I selected those particular trees for a reason!

Factors of 1470:

  • 1470 is a composite number.
  • Prime factorization: 1470 = 2 × 3 × 5 × 7 × 7, which can be written 1470 = 2 × 3 × 5 × 7²
  • 1470 has at least one exponent greater than 1 in its prime factorization so √1470 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1470 = (√49)(√30) = 7√30
  • The exponents in the prime factorization are 1, 1, 1, and 2. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1)(1 + 1)(2 + 1) = 2 × 2 × 2 × 3 = 24. Therefore 1470 has exactly 24 factors.
  • The factors of 1470 are outlined with their factor pair partners in the graphic below.

A Few Factor Trees for 1470:

More Facts about the Number 1470:

1470 is the average of 14² and 14³. That simple fact makes 1470 the 14th Pentagonal Pyramidal Number.

1470 is the hypotenuse of a Pythagorean triple:
882-1176-1470 which is (3-4-5) times 294

Hmm… that same factor pair showed up again!