1512 Bigger Bites of Cake

Using the Cake Method:

I like using the cake method to find the prime factorization of a number. I also use it to find square roots.

If you know the multiplication table well, dividing by any number from 2 to 9 is not difficult.

It is easy to check to see if a number is divisible by 4 or by 9. And it is actually easier to divide by 4 once than it is to divide by 2 twice or to divide by 9 once than it is to divide by 3 twice. That way we get to take bigger bites of cake!

I know that 1512 is divisible by 4 because the number formed from the last two digits in order, 12, is divisible by 4.

I also know that 1512 is divisible by 9 because 1 + 5 + 1 + 2 = 9.

Thus, I’ll begin by dividing first by 4 and then the result by 9 as illustrated below:

When my new divisor becomes 42, if I didn’t remember where it appears in the multiplication table, I would still be fine. I know that 42 is divisible by 2 because it is even and 3 because 4 + 2 = 6, a number divisible by 3.

Thus I can divide 42 by 6 because 2 × 3 = 6, and it is far easier to divide 42 by 6 than it is to divide it first by 2 and then by 3.

To take the square root of 1512, I simply take the square root of the numbers on the outside of the cake. 4 and 9 are perfect squares so I use their square roots. Neither 6 nor 7 has any perfect squares, so I just multiply them together to get 42.

Dividing 1512 this way allowed me to make just a three-layer cake instead of a 6-layer cake to find its prime factorization and its square root!

Some people prefer to do yard work more than having cake, so here is one of the MANY possible factor trees for 1512. Make sure you pick up all seven of the leaves with prime numbers wherever they are.

Factors of 1512:

  • 1512 is a composite number.
  • Prime factorization: 1512 = 2 × 2 × 2 × 3 × 3 × 3 × 7, which can be written 1512 = 2³ × 3³ × 7
  • 1512 has at least one exponent greater than 1 in its prime factorization so √1512 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1512 = (√36)(√42) = 6√42
  • The exponents in the prime factorization are 2, 1, and 2. Adding one to each exponent and multiplying we get (3 + 1)(3 + 1)(1 + 1) = 4 × 4 × 2 = 32. Therefore 1512 has exactly 32 factors.
  • The factors of 1512 are outlined with their factor pair partners in the graphic below.

 

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!

1452 Poinsettia Plant Mystery

Merry Christmas, Everybody!

The poinsettia plant has a reputation for being poisonous, but it has never been a part of a whodunnit, and it never will. Poinsettias actually aren’t poisonous.

Multiplication tables might also have a reputation for being deadly, but they aren’t either, except maybe this one. Can you use logic to solve this puzzle without it killing you?

To solve the puzzle, you will need some multiplication facts that you probably DON’T have memorized. They can be found in the table below. Be careful! The more often a clue appears, the more trouble it can be:

Notice that the number 60 appears EIGHT times in that table. Lucky for you, it doesn’t appear even once in today’s puzzle!

Now I’d like to factor the puzzle number, 1452. Here are a few facts about that number:

1 + 4 + 5 + 2 = 12, which is divisible by 3, so 1452 is divisible by 3.
1 – 4 + 5 – 2 = 0, which is divisible by 11, so 1452 is divisible by 11.

  • 1452 is a composite number.
  • Prime factorization: 1452 = 2 × 2 × 3 × 11 × 11, which can be written 1452 = 2² × 3 × 11²
  • 1452 has at least one exponent greater than 1 in its prime factorization so √1452 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1452 = (√484)(√3) = 22√3
  • The exponents in the prime factorization are 2, 1, and 2. Adding one to each exponent and multiplying we get (2 + 1)(1 + 1)(2 + 1) = 3 × 2 × 3 = 18. Therefore 1452 has exactly 18 factors.
  • The factors of 1452 are outlined with their factor pair partners in the graphic below.

To commemorate the season, here’s a factor tree for 1452:

Have a very happy holiday!

1449 Christmas Star

If you’ve ever wished you knew the multiplication table better, then make that wish upon this Christmas star. If you use logic and don’t give up,  then you can watch your wish come true!

I number the puzzles to distinguish them from one another. That star puzzle is way too big for a factor tree made with its puzzle number:

Here’s more about the number 1449:

  • Prime factorization: 1449 = 3 × 3 × 7 × 23, which can be written 1449 = 3² × 7 × 23
  • 1449 has at least one exponent greater than 1 in its prime factorization so √1449 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1449 = (√9)(√161) = 3√161
  • 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 1449 has exactly 12 factors.
  • The factors of 1449 are outlined with their factor pair partners in the graphic below.

1449 is the difference of two squares in 6 different ways:
725² – 724² = 1449
243² – 240² = 1449
107²-100² = 1449
85² – 76² = 1449
45² – 24² = 1449
43² – 20² = 1449

1428 Factor Trees in Autumn

I recently decided that I wanted to make some factor trees in various fall colors. 1428 has plenty of factors so it has MANY different factor trees. Here are just eleven of them, each initially factored by a different factor pair.

Here are some more facts about the number 1428:

  • 1428 is a composite number.
  • Prime factorization: 1428 = 2 × 2 × 3 × 7 × 17, which can be written 1428 = 2² × 3 × 7 × 17
  • 1428 has at least one exponent greater than 1 in its prime factorization so √1428 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1428 = (√4)(√357) = 2√357
  • The exponents in the prime factorization are 2, 1, 1, and 1. Adding one to each exponent and multiplying we get (2 + 1)(1 + 1)(1 + 1)(1 + 1) = 3 × 2 × 2 × 2 = 24. Therefore 1428 has exactly 24 factors.
  • The factors of 1428 are outlined with their factor pair partners in the graphic below.

1428 is the hypotenuse of a Pythagorean triple:
672-1260-1428 which is (8-15-17) times 84.

1368 Playing with the Sieve of Eratosthenes

What if we didn’t use ten numbers across for our Sieve of Eratosthenes?

For example, 36 × 38 = 1368. We could make a Sieve of Eratosthenes writing 36 numbers across the grid and 38 numbers down. The last number on the grid would be 1368, and we could find all the prime numbers less than 1369 (which is 17²) by crossing out all the multiples of the prime numbers that appear on the top row. The trouble is that 36 numbers across makes a very big grid. Crossing out multiples of 2, 3, 5, and 7 will be very quick, but crossing out all the multiples of 11, 13, 17, 19, 23, 29, and 31 will not be so fun.

Grids that make use of the fact that (n-1)(n+1) = n²-1 can always give us a perfect rectangle and we will only need to cross out the multiples of the prime numbers in the top row to find ALL the prime numbers in any (n-1)×(n+1) list of numbers.

Here’s a 7 × 9 grid:

Since it was 7 across, it was very easy to cross out all the multiples of 7. The multiples of 2 and 3 weren’t too difficult to find either, but the pattern for the multiples of 5 was not quite as nice. Fortunately, it is easy to spot those multiples, no matter how big a number they are.

Still, the first prime number on the second row is 11, so we should be able to go almost up to 11² = 121 on our grid:

I couldn’t fit 120 on the grid without ruining the rectangle, but here’s a grid using 12 numbers across. Since 12 × 14 = 168 which is one less than 13², we can find all the prime numbers in the list simply by crossing out the multiples of the prime numbers in the top row.

But the next number, 13, is only one number more than 12, and all of its multiples are staring at me making me feel very uncomfortable. It will be very easy to cross out all of the multiples of 13. That means we can extend the list of numbers to one less than the next prime number squared, which is 289 – 1 = 288. This time we get a perfect rectangle because 288 is also a multiple of 12:

All of the circled numbers in the top row and every number that has not been crossed out below the top row are prime numbers.

Someone long ago figured out that if we make the grid six numbers across, all the prime numbers except 2 and 3 will appear in the same two columns, no matter how long the grid is:

Every prime number greater than 3 is either one less or one more than a multiple of 6.

Since we always cross out the multiples of 2 anyway, what would happen if we didn’t include them in the grid at all?

Here is a grid with ten numbers across, but only odd numbers are included. Because 5 is a factor of 10, it is very easy to cross out all of the 5’s. Also, since 9 is one less than 10 and 11 is one more than 10, it is also easy to cross out all the multiples of 3 and 11. Crossing out the 7’s and the 19’s wasn’t too bad, either, but the 13’s and 17’s were not as fun.

In my next post, I will share my favorite size of grid and the method I use to find all of the prime numbers on it. No prime numbers get circled in my method.

Some of the numbers in the grids had several lines through them.
If we made the 36 × 38 grid I mentioned at the beginning of the post, how many lines would 1368 have going through it?  After all, 1368 has 24 factors. What do you think?

Only three lines. One each for its prime factors, 2, 3 and 19.

Here’s more about the number 1368:

  • 1368 is a composite number.
  • Prime factorization: 1368 = 2 × 2 × 2 × 3 × 3 × 19, which can be written 1368 = 2³ × 3² × 19
  • 1368 has at least one exponent greater than 1 in its prime factorization so √1368 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1368 = (√36)(√38) = 6√38
  • The exponents in the prime factorization are 3, 2, and 1. Adding one to each exponent and multiplying we get (3 + 1)(2 + 1)(1 + 1) = 4 × 3 × 2 = 24. Therefore 1368 has exactly 24 factors.
  • The factors of 1368 are outlined with their factor pair partners in the graphic below.

Here’s one of the MANY possible factor trees for 1368:

1328 Christmas Tree

 

Can you figure out where to put the numbers from 1 to 10 in both the first column and the top row so that the lights on this Christmas tree work properly?

Print the puzzles or type the solution in this excel file:10-factors-1321-1332

Now I’ll share some facts about the puzzle number, 1328:

  • 1328 is a composite number.
  • Prime factorization: 1328 = 2 × 2 × 2 × 2 × 83, which can be written 1328 = 2⁴ × 83
  • The exponents in the prime factorization are 4 and 1. Adding one to each and multiplying we get (4 + 1)(1 + 1) = 5 × 2 = 10. Therefore 1328 has exactly 10 factors.
  • Factors of 1328: 1, 2, 4, 8, 16, 83, 166, 332, 664, 1328
  • Factor pairs: 1328 = 1 × 1328, 2 × 664, 4 × 332, 8 × 166, or 16 × 83
  • Taking the factor pair with the largest square number factor, we get √1328 = (√16)(√83) = 4√83 ≈ 36.44173

 

Because 28 is divisible by 4, but not by 8, and 3 (the digit before the 28) is an odd number, I know that 1328 is divisible by 8. I can use that fact to make this simple factor tree:

1328 is the difference of two squares three different ways:
333² – 331² = 1328
168² – 164² = 1328
87² – 79²  = 1328

1248 Factor Trees

It is easy to see that 1248 is divisible by 12 and therefore by 2, 3, 4, and 6, but it actually has a lot more factors than that. I decided to celebrate its many factors by making just a few of its many possible factor trees.

Why does 1248 have so many factors? Well, check out its prime factorization:

  • 1248 is a composite number.
  • Prime factorization: 1248 = 2 × 2 × 2 × 2 × 2 × 3 × 13, which can be written 1248 = 2⁵ × 3 × 13
  • The exponents in the prime factorization are 5, 1, and 1. Adding one to each and multiplying we get (5 + 1)(1 + 1)(1 + 1) = 6 × 2 × 2 = 24. Therefore 1248 has exactly 24 factors.
  • Factors of 1248: 1, 2, 3, 4, 6, 8, 12, 13, 16, 24, 26, 32, 39, 48, 52, 78, 96, 104, 156, 208, 312, 416, 624, 1248
  • Factor pairs: 1248 = 1 × 1248, 2 × 624, 3 × 416, 4 × 312, 6 × 208, 8 × 156, 12 × 104, 13 × 96, 16 × 78, 24 × 52, 26 × 48, or 32 × 39
  • Taking the factor pair with the largest square number factor, we get √1248 = (√16)(√78) = 4√78 ≈ 35.32704

1248 is the sum of four consecutive prime numbers:
307 + 311 + 313 + 317 = 1248

1248 is also the hypotenuse of a Pythagorean triple:
480-1152-1248 which is (5-12-13) times 96

How I Knew Immediately that a Factor Pair of 1224 is . . .

12 = 3 × 4 and 24 is one less than 25. Those two facts helped me to know right away that 35² = 1225 and 34 × 36 = 1224. Study the patterns in the chart below and you will likely be able to remember all of the multiplication facts listed in it!

a² – b² = (a – b)(a + b)
You may remember how to factor that from algebra class. Here when b = 1, it has a practical application that can allow you to amaze your friends and family with your mental calculating abilities!

I’ve only typed a small part of that infinite pattern chart. For example, if you know that 19 × 20 = 380, then you can also know that 195² = 38025 and 194 × 196 = 38024.

Also because of that chart, I know that 3.5² = 12.25 and 3.4 × 3.6 = 12.24
(Also (3½)² = 12¼, but 2½  × 4½ = 11¼ because 3-1 = 2, 3+1 = 4, 12-1 = 11
thus 2.5 × 4.5 = 11.25 and 2½  × 4½ = 11¼)

You could also let b = 2 so b² = 4. Then 25 – 4 = 21, and you could know facts like
33 × 37 = 1221 or 193 ×  197 = 38021

I hope you have a wonderful time being a calculating genius!

Now I’ll share some other facts about the number 1224:

  • 1224 is a composite number.
  • Prime factorization: 1224 = 2 × 2 × 2 × 3 × 3 × 17, which can be written 1224 = 2³ × 3² × 17
  • The exponents in the prime factorization are 2, 3 and 1. Adding one to each and multiplying we get (3 + 1)(2 + 1)(1 + 1) = 4 × 3 × 2 = 24. Therefore 1224 has exactly 24 factors.
  • Factors of 1224: 1, 2, 3, 4, 6, 8, 9, 12, 17, 18, 24, 34, 36, 51, 68, 72, 102, 136, 153, 204, 306, 408, 612, 1224
  • Factor pairs: 1224 = 1 × 1224, 2 × 612, 3 × 408, 4 × 306, 6 × 204, 8 × 153, 9 × 136, 12 × 102, 17 × 72, 18 × 68, 24 × 51 or 34 × 36
  • Taking the factor pair with the largest square number factor, we get √1224 = (√36)(√34) = 6√34 ≈ 34.98571

When a number has so many factors, I often will make a forest of factor trees for that number, but today I just want us to enjoy this one tree for 34 × 36 = 1224.

1224 is also the sum of two squares:
30² + 18² = 1224

1224 is the hypotenuse of a Pythagorean triple:
576-1080-1224 which is (8-15-17) times 72
That triple can also be calculated from 30² – 18², 2(30)(18), 30² + 18²

293 + 307 + 311 + 313 = 1224 making 1224 the sum of four consecutive prime numbers.

1170 Factor Trees

1170 is one of those numbers with twenty-four factors. Why does it have so many? Because of its prime factorization. You can find a number’s prime factorization by making a factor tree. Here are eleven different factor trees for 1170. Each one of them produces the same prime factorization: 1170 = 2 × 3² × 5 × 13

  • 1170 is a composite number.
  • Prime factorization: 1170 = 2 × 3 × 3 × 5 × 13, which can be written 1170 = 2 × 3² × 5 × 13
  • The exponents in the prime factorization are 1, 2, 1, and 1. Adding one to each and multiplying we get (1 + 1)(2 + 1)(1 + 1)(1 + 1) = 2 × 3 × 2 × 2 = 24. Therefore 1170 has exactly 24 factors.
  • Factors of 1170: 1, 2, 3, 5, 6, 9, 10, 13, 15, 18, 26, 30, 39, 45, 65, 78, 90, 117, 130, 195, 234, 390, 585, 1170
  • Factor pairs: 1170 = 1 × 1170, 2 × 585, 3 × 390, 5 × 234, 6 × 195, 9 × 130, 10 × 117, 13 × 90, 15 × 78, 18 × 65, 26 × 45, or 30 × 39
  • Taking the factor pair with the largest square number factor, we get √1170 = (√9)(√130) = 3√130 ≈ 34.20526

33² +  9² = 1170
27² +  21² = 1170

1170 is also the hypotenuse of FOUR Pythagorean triples:
288-1134-1170 calculated from 27² –  21², 2(27)(21), 27² +  21²
450-1080-1170 which is (5-12-13) times 90
594-1008-1170 calculated from 2(33)( 9), 33² –  9², 33² +  9²
702-936-1170 which is (3-4-5) times 234

1170 is 102102 in BASE 4 because 4⁵ + 2(4³) +4² + 2(1) = 1170,
and it’s repdigit 2222 in BASE 8 because 2(8³ + 8² + 8¹ + 8⁰) = 2(585) = 1170