1696 Inverses

Today’s Puzzle:

Joseph Nebus of Nebusresearch recently wrote an extensive post about inverses at my request. Although inverses are often fascinating, advanced topics in mathematics, they can also be quite simple. For example, solving this puzzle will involve using the inverse of multiplication, division, AND it is the simplest division possible this time. Since it is December, I made this puzzle look like a gift for you. Just write the numbers 1 to 12 in both the first column and the top row so that those numbers and the given clues could become a multiplication table.

For any gift you receive, you can invoke inverse exploring questions such as “how was this gift made?”

If you want to print the puzzle without color, here it is:

Factors of 1696:

  • 1696 is a composite number.
  • Prime factorization: 1696 = 2 × 2 × 2 × 2 × 2 × 53, which can be written 1696 = 2⁵ × 53.
  • 1696 has at least one exponent greater than 1 in its prime factorization so √1696 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1696 = (√16)(√106) = 4√106.
  • The exponents in the prime factorization are 5 and 1. Adding one to each exponent and multiplying we get (5 + 1)(1 + 1) = 6 × 2 = 12. Therefore 1696 has exactly 12 factors.
  • The factors of 1696 are outlined with their factor pair partners in the graphic below.

More About the Number 1696:

1696 is the sum of two squares:
36² + 20² = 1696.

1696 is also the hypotenuse of a Pythagorean triple:
896-1440 -1696, which is 32 times (28-45-52).
It can also be calculated from 36² – 20², 2(36)(20), 36² + 20².

1696 is also the difference of two squares in FOUR different ways:
425² – 423² = 1696,
214² – 210² = 1696,
110² – 102² = 1696, and
61² – 45² = 1696.
I found those equations by using the even factor pairs of 1696 and taking the inverse of the fact that a² – b² = (a + b)(a – b).

1683 Grave Marker

Today’s Puzzle:

It’s almost Halloween. I hope you enjoy this grave-marker puzzle. Write the numbers from 1 to 12 in both the first column and the top row so that those numbers and the given clues make a multiplication table.

Here’s the same puzzle, but it won’t use up all your printer ink.

Factors of 1683:

  • 1683 is a composite number.
  • Prime factorization: 1683 = 3 × 3 × 11 × 17, which can be written 1683 = 3² × 11 × 17.
  • 1683 has at least one exponent greater than 1 in its prime factorization so √1683 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1683 = (√9)(√187) = 3√187.
  • 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 1683 has exactly 12 factors.
  • The factors of 1683 are outlined with their factor pair partners in the graphic below.

More About the Number 1683:

1683 is the hypotenuse of a Pythagorean triple:
792-1485-1683, which is (8-15-17) times 99.

1683 is the difference of two squares in SIX different ways:
842² – 841² = 1683,
282² – 279² = 1683,
98² – 89² = 1683,
82² – 71² = 1683,
58² – 41² = 1683, and
42² – 9² = 1683.
That last one means we are 81 numbers away from the next perfect square. I also highlighted a cool-looking difference.

1680, 1681, 1682, 1683, and 1684 are the second smallest set of FIVE consecutive numbers whose square roots can be simplified.

1680 square roots

1673 and Level 1

Today’s Puzzle:

Write the numbers from 1 to 10 in both the first column and the top row so that those numbers and the given clues become a not-in-the-usual-order multiplication table.

Factors of 1673:

  • 1673 is a composite number.
  • Prime factorization: 1673 = 7 × 239.
  • 1673 has no exponents greater than 1 in its prime factorization, so √1673 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 1673 has exactly 4 factors.
  • The factors of 1673 are outlined with their factor pair partners in the graphic below.

More About the Number 1673:

The factors of 1673 are 1, 7, 239, and 1673. OEIS.org tells us something cool about the sum of the squares of those factors:
1² + 7² + 239² + 1673² = 1690².

1663 and Level 1

Today’s Puzzle:

Write the numbers 1 to 12 in both the first column and the top row so that those numbers and the given clues function like a multiplication table.

Factors of 1663:

  • 1663 is a prime number.
  • Prime factorization: 1663 is prime.
  • 1663 has no exponents greater than 1 in its prime factorization, so √1663 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1663 has exactly 2 factors.
  • The factors of 1663 are outlined with their factor pair partners in the graphic below.

How do we know that 1663 is a prime number? If 1663 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1663. Since 1663 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, or 37, we know that 1663 is a prime number.

More About the Number 1663:

1663 is the sum of consecutive numbers in only one way:
831 + 832 = 1663.

1663 is the difference of two squares in only one way:
832² – 831² = 1663.

What do you notice about those two number facts?

1650 Wrinkles in the Multiplication Table

Today’s Puzzle:

Are you familiar with the book A Wrinkle in Time? Kat of The Lily Cafe’s blog loves books and recently compared Meg in that book to her six-year-old son. She wrote a post titled Am I Raising a Meg? Her six-year-old LOVES math and is very much interested in multiplication and division. When Mom thought he was playing a game on her phone, he was actually playing with the calculator app! I felt so happy inside as I read that!

I wonder if they have discovered the storybooks in the Math Book Magic blog. Such books could combine Mom’s love for reading with her son’s love of math.

Someday her son might like to solve a “wrinkled” multiplication table puzzle like this one that has only nine clues.

Write all the numbers 1 to 10 in both the first column and the top row so that those numbers and the given clues become a multiplication table.

Factor Cake for 1650:

This is my 1650th post.
1650 is divisible by 2 and by 5 because it ends with a 0.
1650 is divisible by 3 because 1 + 6 + 5 + 0 = 12, a number divisible by 3.
1650 is divisible by 11 because 1 – 6 + 5 – 0 = 0, a number divisible by 11.

I think we can make a lovely factor cake for 1650:

Factors of 1650:

  • 1650 is a composite number.
  • Prime factorization: 1650 = 2 × 3 × 5 × 5 × 11, which can be written 1650 = 2 × 3 × 5² × 11.
  • 1650 has at least one exponent greater than 1 in its prime factorization so √1650 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1650 = (√25)(√66) = 5√66.
  • The exponents in the prime factorization are 1, 1, 2, and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1)(2 + 1)(1 + 1) = 2 × 2 × 3 × 2 = 24. Therefore 1650 has exactly 24 factors.
  • The factors of 1650 are outlined with their factor pair partners in the graphic below.

More About the Number 1650:

1650 is the hypotenuse of TWO Pythagorean triples:
462-1584-1650, which is (7-24-25) times 66, and
990-1320-1650, which is (3-4-5) times 330.

1639 and Level 1

Today’s Puzzle:

Write the numbers from 1 to 12 in both the first column and the top row so that those numbers and the given clues will make this puzzle function like a multiplication table.

Factors of 1639:

1 – 6 + 3 – 9 = -11 so 1639 is divisible by 11.

  • 1639 is a composite number.
  • Prime factorization: 1639 = 11 × 149.
  • 1639 has no exponents greater than 1 in its prime factorization, so √1639 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 1639 has exactly 4 factors.
  • The factors of 1639 are outlined with their factor pair partners in the graphic below.

More about the Number 1639:

1639 is the hypotenuse of a Pythagorean triple:
561 1540 1639, which is 11 times (51-140-149).

1639 is the 22nd nonagonal number because
22(7·22 – 5)/2 =
22(154 – 5)/2=
22(149)/2 =
11(149) = 1639.
Mathworld.Wolfram has illustrations of the first 5 nonagonal numbers.

1628 A Simple Cross

Today’s Puzzle:

A simple cross is an appropriate symbol for Good Friday. Write the numbers from 1 to 10 in the first column and the top row so that those numbers and the given clues function like a multiplication table.

Factors of 1628:

  • 1628 is a composite number.
  • Prime factorization: 1628 = 2 × 2 × 11 × 37, which can be written 1628 = 2² × 11 × 37.
  • 1628 has at least one exponent greater than 1 in its prime factorization so √1628 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1628 = (√4)(√407) = 2√407.
  • 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 1628 has exactly 12 factors.
  • The factors of 1628 are outlined with their factor pair partners in the graphic below.

More about the Number 1628:

1628 is the hypotenuse of a Pythagorean triple:
528-1540-1628, which is (12-35-37) times 44.

1628 is the difference of two squares in two different ways:
408² – 406² = 1628, and
48² – 26²  = 1628.

1605 and Another Big Level One

Today’s Puzzle:

Many of the clues in this puzzle do not appear in a 1 to 10 puzzle or a 1 to 12 puzzle. Can you find all the factors anyway? Solving this puzzle can help you solve any other Find the Factors 1 -14 puzzles.

Print the puzzles or type the solution in this excel file: 14 Factors 1604-1612.

Factors of 1605:

  • 1605 is a composite number.
  • Prime factorization: 1605 = 3 × 5 × 107.
  • 1605 has no exponents greater than 1 in its prime factorization, so √1605 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 1605 has exactly 8 factors.
  • The factors of 1605 are outlined with their factor pair partners in the graphic below.

More about the Number 1605:

1605 is the hypotenuse of a Pythagorean triple:
963-1284-1605, which is (3-4-5) times 321.

1605 is the difference of two squares four different ways:
803² – 802² = 1605,
269² – 266² = 1605,
163² – 158² = 1605, and
61² – 46² = 1605.

1604 and a Big Level One

Today’s Puzzle:

1604 was my house number most of my childhood. Perhaps this number is making me feel a little more playful. I decided my next few puzzles would be based on a 1 to 14 multiplication table. I find that I personally use clues like 26, 39, 52, 65, 78, 104, 143, and 169 often in life. For example, why are there 52 playing cards in a deck of cards? To help you get more familiar with the 1 to14 multiplication table, we will start with this level one puzzle:

Print the puzzles or type the solution in this excel file: 14 Factors 1604-1612.

Factors of 1604:

  • 1604 is a composite number.
  • Prime factorization: 1604 = 2 × 2 × 401, which can be written 1604 = 2² × 401.
  • 1604 has at least one exponent greater than 1 in its prime factorization so √1604 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1604 = (√4)(√401) = 2√401.
  • 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 1604 has exactly 6 factors.
  • The factors of 1604 are outlined with their factor pair partners in the graphic below.

More about the Number 1604:

1604 is the sum of two squares:
40² + 2² = 1604.

1604 is the hypotenuse of a Pythagorean triple:
160-1596-1604, calculated from 2(40)(2), 40² – 2², 40² + 2².
It is also 4 times (40-399-401).

1594 and Level 1

Today’s Puzzle:

Write the numbers from 1 to 10 in the first column and the top row so that those numbers and the given clues function like a multiplication table.

Factors of 1594:

  • 1594 is a composite number.
  • Prime factorization: 1594 = 2 × 797.
  • 1594 has no exponents greater than 1 in its prime factorization, so √1594 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 1594 has exactly 4 factors.
  • The factors of 1594 are outlined with their factor pair partners in the graphic below.

More about the Number 1594:

1594 is the sum of two squares as well as the double of two squares:
37² + 15² = 1594.
2(26² + 11²) = 1594.

1594 is the hypotenuse of a Pythagorean triple:
1110-1144-1594, calculated from 2(37)(15), 37² – 15², 37² + 15².
It is also 2 times (555-572-797).