1474 and Level 6

Today’s Puzzle:

If you think that only kids will find the Find the Factors puzzles a challenge, then this puzzle will likely change your mind. Seriously, can you use logic to find where the factors from 1 to 12 must go to complete this puzzle? There is a reason this is a level 6 puzzle!

Factors of 1474:

The last digit of 1474 is even so 1474 is divisible by 2. Yes, that was easy, but try this next one:
1 – 4 + 7 – 4 = 0, so 1474 can be evenly divided by 11.

  • 1474 is a composite number.
  • Prime factorization: 1474 = 2 × 11 × 67
  • 1474 has no exponents greater than 1 in its prime factorization, so √1474 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 1474 has exactly 8 factors.
  • The factors of 1474 are outlined with their factor pair partners in the graphic below.

Another Fact about the Number 1474:

1474 can not be written as the sum of two squares or the difference of two squares, but it is still the short leg in two Pythagorean triples. Here is one of them:
1474-4368-4610, calculated from 2(67)(11), 67² – 11², 67² + 11²

 

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1473 A Level 5 Designed by My Granddaughter

Today’s Puzzle:

My son and his family arrived today for a two-month long visit. We are so excited that they are here. His nine-year-old daughter has solved her share of Find the Factors puzzles. Today I gave her the opportunity to help me design one which she thoroughly loved doing. Here is the first of her creations. Will you be able to solve it? It’s a level 5 so you might find it a bit tricky.

Factors of 1473:

  • 1473 is a composite number.
  • Prime factorization: 1473 = 3 × 491
  • 1473 has no exponents greater than 1 in its prime factorization, so √1473 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 1473 has exactly 4 factors.
  • The factors of 1473 are outlined with their factor pair partners in the graphic below.

More about the number 1473:

1473 can be written as the difference of two squares in two different ways:
737² – 736² = 1473
247² – 244² = 1473

1472 and Level 4

Today’s Puzzle:

Can you use logic to figure out where the factors from 1 to 12 must go to make the given clues be the products of those factors?

Factors of 1472:

  • 1472 is a composite number.
  • Prime factorization: 1472 = 2 × 2 × 2 × 2 × 2 × 2 × 23, which can be written 1472 = 2⁶ × 23.
  • 1472 has at least one exponent greater than 1 in its prime factorization so √1472 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1472 = (√64)(√23) = 8√23
  • The exponents in the prime factorization are 6 and 1. Adding one to each exponent and multiplying we get (6 + 1)(1 + 1) = 7 × 2 = 14. Therefore 1472 has exactly 14 factors.
  • The factors of 1472 are outlined with their factor pair partners in the graphic below.

A Little More about the Number 1472:

1472 is the difference of two squares in FIVE different ways:
369² – 367² = 1472
186² – 182² = 1472
96² – 88² = 1472
54² – 38² = 1472
39² – 7² = 1472

 

1471 and Level 3

Today’s Puzzle:

There is only one factor that will work with the clue in the top row of this level 3 puzzle. Find it, then work your way down row by row until you have found all the factors. There is only one solution.

Factors of 147

  • 1471 is a prime number.
  • Prime factorization: 1471 is prime.
  • 1471 has no exponents greater than 1 in its prime factorization, so √1471 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1471 has exactly 2 factors.
  • The factors of 1471 are outlined with their factor pair partners in the graphic below.

How do we know that 1471 is a prime number? If 1471 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1471. Since 1471 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 31, or 37, we know that 1471 is a prime number.

Another Fact about the Number 1471:

1471 is the 21st centered heptagonal number because
(7(21²)-7(21)+2)/2 = 1471.

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!

1469 and Level 2

Today’s Puzzle:

Where do the Factors 1-12 belong on this multiplication table puzzle?

Factors of 1469:

  • 1469 is a composite number.
  • Prime factorization: 1469 = 13 × 113
  • 1469 has no exponents greater than 1 in its prime factorization, so √1469 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 1469 has exactly 4 factors.
  • The factors of 1469 are outlined with their factor pair partners in the graphic below.

Other Facts about the Number 1469:

1469 is the sum of two squares in two different ways:
38² + 5² = 1469
37² + 10² = 1469

1469 is the hypotenuse of FOUR Pythagorean triples:
195-1456-1469 which is 13 times (15-112-113)
380-1419-1469 calculated from 2(38)(5), 38² – 5², 38² + 5²
565-1356-1469 which is (5-12-13) times 113
740-1269-1469 calculated from 2(37)(10), 37² – 10², 37² + 10²

 

1468 and Level 1

Today’s Puzzle

Can you find the factors from 1 to 12 that will work with the given clues to turn this puzzle into a multiplication table?


Factors of 1468

  • 1468 is a composite number.
  • Prime factorization: 1468 = 2 × 2 × 367, which can be written 1468 = 2² × 367
  • 1468 has at least one exponent greater than 1 in its prime factorization so √1468 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1468 = (√4)(√367) = 2√367
  • 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 1468 has exactly 6 factors.
  • The factors of 1468 are outlined with their factor pair partners in the graphic below.

Another Interesting Fact about the number 1468:

Stetson.edu informs us that the smallest 20-digit 6th power is 1468⁶.

1466 A Non-Simplified Yet Easy-to-Remember Unit Circle

Look at the Patterns in this Unit Circle:

A unit circle has a radius of one and is often used to display the sines and cosines of special angles. I’ve made a unit circle that showcases the patterns that make memorizing how to complete one easier.

This unit circle is easy to memorize because many of the values have not been simplified. However, always remember to simplify the values when giving your final answer on any assignment or test!

The patterns you see in the ordered pairs representing the cosines and sines are only coincidental: they are merely an aid somebody else created to make memorizing the values easier. Over half of the square roots can be simplified and should be. However, I do love them in their non-simplified form because it is easy to remember them, and it is easy to see the sine wave and the cosine wave as the square root values increase and decrease incrementally as we go around the unit circle.

I’ve also made a similar unit circle that might help students locate the angles in the quadrants better:

Factors of 1466:

  • 1466 is a composite number.
  • Prime factorization: 1466 = 2 × 733
  • 1466 has no exponents greater than 1 in its prime factorization, so √1466 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 1466 has exactly 4 factors.
  • The factors of 1466 are outlined with their factor pair partners in the graphic below.

More Facts about the number 1466:

1466 is the sum of two squares:
29² + 25² = 1466

1466 is also the hypotenuse of a Pythagorean triple:
216-1450-1466 calculated from 29² – 25², 2(29)(25), 29² + 25²

1465 A Degrees to and from Radians Calculator

Degrees vs. Radians

Elementary and middle school students measure their angles in degrees. When they get to high school, they find out that sometimes we prefer to measure angles in radians. Of course, they also need to convert degrees to radians and to convert radians into degrees.

Degrees to/from Radians Calculator:

Since I have students who rely on calculators but don’t have access to one during this pandemic, I designed a calculator in Desmos to convert Degrees to and from Radians. The calculator doesn’t work like magic, still requires you to think about what you are doing, and should prevent some of the common errors in doing such conversions.

Factors of 1465:

  • 1465 is a composite number.
  • Prime factorization: 1465 = 5 × 293
  • 1465 has no exponents greater than 1 in its prime factorization, so √1465 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 1465 has exactly 4 factors.
  • The factors of 1465 are outlined with their factor pair partners in the graphic below.

More Facts about the Number 1465:

1465 is the sum of two squares in two different ways:
36² + 13² = 1465
32² + 21² = 1465

1465 is also the hypotenuse of FOUR Pythagorean triples:
340-1425-1465 which is 5 times 68-285-293.
583-1344-1465 calculated from 36² – 13², 2(36)(13), 36² + 13².
879-1172-1465 which is (3-4-5) times 293.
936-1127-1465 calculated from 2(36)(13), 36² – 13², 36² + 13².

From Stetson.edu, we learn this fun square number fact:
1465² = 2146225.

1464 Alibis and a Mystery

Today’s Puzzle:

You find clues 30, 30, 30, 54. Two of those 30’s claim to be 3 × 10.
When you find clues 40, 40, 56, you realize that at least one of the 40’s must be 4 x 10.

Can you believe the alibis the 30’s have just given you?
Can you put your detective skills together to figure out this mystery?

Factors of 1464:

  • 1464 is a composite number.
  • Prime factorization: 1464 = 2 × 2 × 2 × 3 × 61, which can be written 1464 = 2³ × 3 × 61
  • 1464 has at least one exponent greater than 1 in its prime factorization so √1464 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1464 = (√4)(√366) = 2√366
  • 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 1464 has exactly 16 factors.
  • The factors of 1464 are outlined with their factor pair partners in the graphic below.

More Facts about the Number 1464:

1464 is the hypotenuse of a Pythagorean triple:
264-1440-1464 which is 24 times (11-60-61).

Stetson.edu alerts us that 1464 is a repdigit in two different bases:
It’s 1111 in BASE 11 because 11³ + 11² + 11¹ + 11º = 1464, and
it’s 888 in BASE 13 because 8(13² + 13¹ + 13º) = 8(183) =1464.