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).

1586 and a Level One Puzzle

Today’s Puzzle:

Write the number from 1 to 12 in both the first column and the top row of the puzzle so that the given clues are the products of the numbers you write.

Factors of 1586:

  • 1586 is a composite number.
  • Prime factorization: 1586 = 2 × 13 × 61.
  • 1586 has no exponents greater than 1 in its prime factorization, so √1586 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 1586 has exactly 8 factors.
  • The factors of 1586 are outlined with their factor pair partners in the graphic below.

More about the Number 1586:

1586 is the sum of two squares in two different ways:
35² + 19² = 1586, and
31² + 25² = 1586.

1586 is the hypotenuse of FOUR Pythagorean triples:
286-1560-1586, which is 26 times (11-60-61),
336-1550-1586, calculated from 31² – 25², 2(31)(25), 31² + 25²,
610-1464-1586, which is (5-12-13) times 122, and
864-1330-1586, calculated from 31² – 25², 2(31)(25), 31² + 25².

1576 Why Do the Digits of Every Clue in This Puzzle Add up to Nine?

Today’s Puzzle:

The digits of each clue in this puzzle add up to nine. Relatively few Find the Factors puzzles can make that claim. Why can this one make it?

Can you write all the numbers from 1 to 10 so that this puzzle will function like a multiplication table?

Factors of 1576:

1576 is divisible by a few powers of two, namely 1, 2, 4, and 8. Here is something for you to think about:

1576 is a whole number so it is divisible by 1.
6 is even so 1576 is divisible by 2.
7 is odd and 6 is not divisible by 4, so 1576 is divisible by 4.
5 is odd and 76 is not divisible by 8, so 1576 is divisible by 8.
1 is odd and 576 is divisible by 16, so 1576 is not divisible by 16.
(Each of those statements was also dependent on the statement before it.)

  • 1576 is a composite number.
  • Prime factorization: 1576 = 2 × 2 × 2 × 197, which can be written 1576 = 2³ × 197.
  • 1576 has at least one exponent greater than 1 in its prime factorization so √1576 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1576 = (√4)(√394) = 2√394.
  • 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 1576 has exactly 8 factors.
  • The factors of 1576 are outlined with their factor pair partners in the graphic below.

More about the Number 1576:

1576 is the sum of two squares:
30² + 26² = 1576.

1576 is the hypotenuse of a Pythagorean triple:
224-1560-1576, calculated from 30² – 26², 2(30)(26), 30² + 26².
It is also 8 times (28-195-197).

1576 is also the difference of two squares in two different ways:
395² – 393² = 1576 and
199² – 195² = 1576.