Prime factorization

1646 Mystery Level

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Today’s Puzzle:

It’s a mystery if this puzzle is easy, difficult, or half easy and half difficult. The only way for you to know is to use logic to try solving it yourself!

Factors of 1646:

  • 1646 is a composite number.
  • Prime factorization: 1646 = 2 × 823.
  • 1646 has no exponents greater than 1 in its prime factorization, so √1646 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 1646 has exactly 4 factors.
  • The factors of 1646 are outlined with their factor pair partners in the graphic below.

More About the Number 1646:

823 is part of the prime decade: 821, 823, 827, 829.
You can be sure that each of those primes doubled: 1642, 1646, 1654, and 1658 will have exactly four factors.

1646 is in only one Pythagorean triple:
1646-677328-677330, calculated from 2(823)(1), 823² – 1², 823² + 1².

1645 and Level 6

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Today’s Puzzle:

The number 36 appears as a clue in this puzzle three times. None of those 36’s will be 3 × 12 because 21 and 33 must use both 3’s. That means two of the 36’s will be 4 × 9, and one of them will be 6 × 6. Can both of the 36’s associated with the 24 be 4 × 9? Answering that question will help you find the logic needed to know which common factor to use for 72 and 36.

Factors of 1645:

  • 1645 is a composite number.
  • Prime factorization: 1645 = 5 × 7 × 47.
  • 1645 has no exponents greater than 1 in its prime factorization, so √1645 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 1645 has exactly 8 factors.
  • The factors of 1645 are outlined with their factor pair partners in the graphic below.

 

More About the Number 1645:

1645 is the hypotenuse of a Pythagorean triple:
987-1316-1645, which is (3-4-5) times 329.

1644 Level 5 Puzzles Are Not So Easy to Solve

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Today’s Puzzle:

This puzzle isn’t so easy to solve. For example, the common factor of clues 48 and 24 might be 4, 6, 8, or 12. Which one should you use? Logic will help answer that question. Give this puzzle a try?

Factors of 1644:

1644 has 12 factors and is divisible by 12.

  • 1644 is a composite number.
  • Prime factorization: 1644 = 2 × 2 × 3 × 137, which can be written 1644 = 2² × 3 × 137.
  • 1644 has at least one exponent greater than 1 in its prime factorization so √1644 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1644 = (√4)(√411) = 2√411.
  • 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 1644 has exactly 12 factors.
  • The factors of 1644 are outlined with their factor pair partners in the graphic below.

More about the Number 1644:

1644 is the hypotenuse of a Pythagorean triple:
1056-1260-1644 which is 12 times (88-105-137).

1643 Thrift Store Find

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Today’s Puzzle:

Yesterday I walked to a thrift store located about a mile and a half from my house. I looked through their picture books and was delighted to find all of these:

I love reading math picture books to my grandchildren. My eighteenth (nine girls and nine boys) grandchild was born earlier this month, so each book will be treasured.

The display of books made me think of some math questions:

How many books do you see? How did you count them?

How many of the books have you read? How many haven’t you read?

If you’ve read X number of the books, but haven’t read Y of them, write an expression for the total number of books.

The thrift store price of each book was $1.29. How much did I expect to pay for all these books?

I wasn’t aware of the store’s “buy four get one free” policy before I checked out. How many free books did I get? Could I have done better than that?

I was charged 7.25% sales tax. What was the total tax I paid? What was the total amount I paid for the books?

I didn’t have time to look through every shelf of books yesterday, so today I went back and found a few more treasures. (I also bought four non-mathy books that are not pictured.) Ten Apples Up On Top was a book I read to my children when they were young. We loved it so much that eventually, it fell apart. Some of my grandchildren haveTen Tiny Tickles in their home libraries, but now they can enjoy it at my house as well.

What kind of math can be found in the books I bought?

  1. One Hundred Days (Plus One) – What number comes after 100?
  2. The Stephen Cartwright 123 – Count to 21.
  3. Boom Chicka Rock – Hour hand clock math.
  4. Numbears – Count to 12.
  5. Ten Happy Whales – Adding one more; Introduction to addition.
  6. 10 Black Dots – Count to 10.
  7. One Hundred Hungry Ants – Dividing 100 by 2, 4, 5, and 10.
  8. A Remainder of One – Dividing 25 by 2, 3, or 4 leaves a remainder of one, but dividing by 5 does not.
  9. Sixteen Runaway Pumpkins – Powers of 2.
  10. 10 for dinner – Ways to add four or five numbers to make 10.
  11. Great Estimations – Estimating tens, hundreds, or thousands of objects.
  12. Baby Counts – Count to 4.
  13. Counting Farm – Count to 10.
  14. 123 First Board Book – Count to 10, 20, 50, 100.
  15. Ten Apples Up On Top – Count to 10 (Count by 10’s also).
  16. Cookie’s Week – Calendar math – Days of the week.
  17. Millions of Snowflakes – Count to 5.
  18. A House for Hermit Crab – Calendar Math – Months of the Year.
  19. Ten Tiny Tickles – Count to 10.
  20. Each Orange Had 8 Slices – Word problems, multiplying three numbers together.
  21. One Less Fish – Counting down from 12 to 0.
  22. Lunch Money and Other Poems about School has three poems about math: Eight-Oh-Three (clock math), Lunch Money (types of coins), Math My Way (should 2 + 2 be 22 instead of 4? What about 3 + 3 and 4 + 4?)
  23. Nine O’Clock Lullaby – Clock math, time zones around the world.
  24. One Monday Morning – Calendar Math, Days of the week.
  25. A Quarter from the Tooth Fairy – Different ways to make 25¢.

All of these books have charming pictures, and I was confident I would enjoy reading them aloud. So many books, so little time to spend with grandkids! We didn’t read even half of them today, but we enjoyed the ones we did read very much. I will mention only three of them here:

  1. Baby Counts has only ten words in it, but the pictures made my 2-year-old grandson laugh so much that we reread it as soon as we finished it several times. He already knew how to count to four and delighted in counting along with me.
  2. Ten Apples Up On Top is as captivating now as it was when my children were young.
  3. I was surprised how much kids who have never heard of the tooth fairy enjoyed A Quarter from the Tooth Fairy.

I look forward to enjoying all of the books with my grandkids when they visit me in the future.

I looked but didn’t find any math-related games at this thrift store this time. One of the people I follow on Twitter reminds us that some great games can often be found there.

Factors of 1643:

  • 1643 is a composite number.
  • Prime factorization: 1643 = 31 × 53.
  • 1643 has no exponents greater than 1 in its prime factorization, so √1643 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 1643 has exactly 4 factors.
  • The factors of 1643 are outlined with their factor pair partners in the graphic below.

More about the number 1643:

1643 is the hypotenuse of a Pythagorean triple:
868-1395-1643, which is 31 times (28-45-53).

From OEIS.org, we learn that not only is
1643 = 31 × 53, but also
1643₁₀ = 3153₈.

1642 and Level 4

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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).

1641 and Level 3

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Today’s Puzzle:

This is a level 3 puzzle so the clues have been placed so that you can know what order you should use the clues. Place the factors of 90 and 30 in the appropriate cells, then work your way down the puzzle cell by cell filling in the factors of the clues as you go.

Factors of 1641:

1 + 4 + 1 = 6, so 1641 is divisible by 3. (It isn’t necessary to include multiples of 3 in the sum to determine divisibility by 3.)

  • 1641 is a composite number.
  • Prime factorization: 1641 = 3 × 547.
  • 1641 has no exponents greater than 1 in its prime factorization, so √1641 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 1641 has exactly 4 factors.
  • The factors of 1641 are outlined with their factor pair partners in the graphic below.

More About the Number 1641:

1641 is the difference of two squares in two different ways:
821² – 820² = 1641, and
275² – 272² = 1641.

From OEIS.org we learn that the number formed from 1²6²4²1² is a perfect square:
Recall that 1² = 1; 6² = 36; 4² = 16; and 1² = 1. Those squares form the number, 136161.
Sure enough, 136161 = 369².

1640 A Level 2 Flower

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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 function like a multiplication table.

Factors of 1640:

  • 1640 is a composite number.
  • Prime factorization: 1640 = 2 × 2 × 2 × 5 × 41, which can be written 1640 = 2³ × 5 × 41.
  • 1640 has at least one exponent greater than 1 in its prime factorization so √1640 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1640 = (√4)(√410) = 2√410.
  • 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 1640 has exactly 16 factors.
  • The factors of 1640 are outlined with their factor pair partners in the graphic below.

More about the Number 1640:

Since 1640 = 40 x 41, we can be sure that 1640 is the sum of the first 40 even numbers.

1640 is the sum of two squares in two different ways:
38² + 14² = 1640, and
34² + 22² = 1640.

1640 is the hypotenuse of a Pythagorean triple in FOUR different ways:
360-1600-1640, which is 40 times (9-40-41),
672-1496-1640, calculated from 34² – 22², 2(34)(22), 34² + 22²,
but it is also 8 times (84-187-205),
984-1312-1640, which is (3-4-5) times 328, and
1064-1248-1640, calculated from 2(38)(14), 38² – 14², 38² + 14²,
but it is also 8 times (133-156-205).

1640₁₀ = 2222₉ because 2(9³ + 9² + 9¹ + 9⁰) = 2(729 + 81 + 9 + 1) = 2(820) = 1640.
1640₁₀ = 2020202₃ because 2(3⁶ + 3⁴ + 3² + 3⁰) = 2(729 + 81 + 9 + 1) = 2(820) = 1640.

 

1639 and Level 1

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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.

1638 Factors and Multiples

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Mathematical Musings:

I like the way this tweet shows familiar relationships of several unfamiliar math terms.

Recalling that MANY people confuse factors with multiples, I was inspired to make something similar that will hopefully help people to know which is which:

Factors of 1638:

1638 is even, so it is divisible by 2.
1 + 8 = 9 and 6 + 3 = 9, so 1638 is divisible by 9.

  • 1638 is a composite number.
  • Prime factorization: 1638 = 2 × 3 × 3 × 7 × 13, which can be written 1638 = 2 × 3² × 7 × 13.
  • 1638 has at least one exponent greater than 1 in its prime factorization so √1638 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1638 = (√9)(√182) = 3√182.
  • The exponents in the prime factorization are 1, 2, 1, and 1. Adding one to each exponent and multiplying we get (1 + 1)(2 + 1)(1 + 1)(1 + 1) = 2 × 3 × 2 × 2 = 24. Therefore 1638 has exactly 24 factors.
  • The factors of 1638 are outlined with their factor pair partners in the graphic below.

More about the Number 1638 and Today’s Puzzle:

1638₁₀ = 666₁₆ because 6(16² + 16¹ + 16º) = 6(256 + 16 + 1) = 6(273) = 1638.

1638 is the hypotenuse of a Pythagorean triple:
630-1512-1638.

Solution:
It may be a little confusing because of the words greatest and least, but remember factor ≤ multiple so
Greatest Common Factor < Least Common Multiple.
Or simply, GCF < LCM.

Since these numbers have several digits, skip counting to find common multiples isn’t practical. An easy way to solve the puzzle is to pay attention to the exponents in the prime factorizations:
630 = 2 × 3² × 5 × 7,
1512 = 2³ × 3³ × 7, and
1638 = 2 × 3² × 7 × 13.

Rewrite the prime factorizations to contain all the bases used in any of the prime factorizations with the appropriate exponents:
630 = 2¹ × 3² × 5¹ × 7¹ × 13º,
1512 = 2³ × 3³ × 5º × 7¹ × 13º, and
1638 = 2¹ × 3² × 5º × 7¹ × 13¹.

Write the bases using the SMALLEST exponents for the Greatest Common Factor:
GCF = 2¹ × 3² × 5º × 7¹ × 13º = 2 × 3² × 7 = 126.

Write the bases using the LARGEST exponents for Least Common Multiple:
LCM = 2³ × 3³ × 5¹ × 7¹ × 13¹ = 98280.
(Aren’t you glad we didn’t skip count to find it!)

Is GCF < LCM?
126 < 98280. Most certainly!

As you might expect, 630-1512-1638 is 126 times (5-12-13).

 

1637 Flower Challenge

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Today’s Puzzle:

Mother’s Day in the United States is this Sunday so I made this challenging flower puzzle for the occasion.

Use logic to write the numbers from 1 to 10 in each of the four boldly outlined areas so that those numbers and the given clues work together to make four multiplication tables.

Factors of 1637:

  • 1637 is a prime number.
  • Prime factorization: 1637 is prime.
  • 1637 has no exponents greater than 1 in its prime factorization, so √1637 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1637 has exactly 2 factors.
  • The factors of 1637 are outlined with their factor pair partners in the graphic below.

How do we know that 1637 is a prime number? If 1637 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1637. Since 1637 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, or 37, we know that 1637 is a prime number.

More about the Number 1637:

1637 is the sum of two squares:
31² + 26² = 1637.

1637 is the hypotenuse of a Pythagorean triple:
285-1612-1637, calculated from 31² – 26², 2(31)(26), 31² + 26².

Here’s another way we know that 1637 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 31² + 26² = 1637 with 31 and 26 having no common prime factors, 1637 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1637. Since 1637 is not divisible by 5, 13, 17, 29, or 37, we know that 1637 is a prime number.

Do you notice anything else special about the number 1637 in this color-coded chart?