A Multiplication Based Logic Puzzle

827 and Level 5

827 is one of the prime numbers in the fourth prime decade, (821, 823, 827, 829).

827 = 103 + 107+ 109 + 113+ 127 + 131 + 137, that’s the sum of 7 consecutive prime numbers.

Here’s today’s puzzle:

Print the puzzles or type the solution on this excel file: 10-factors-822-828

  • 827 is a prime number.
  • Prime factorization: 827 is prime.
  • The exponent of prime number 827 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 827 has exactly 2 factors.
  • Factors of 827: 1, 827
  • Factor pairs: 827 = 1 × 827
  • 827 has no square factors that allow its square root to be simplified. √827 ≈ 28.7576

How do we know that 827 is a prime number? If 827 were not a prime number, then it would be divisible by at least one prime number less than or equal to √827 ≈ 28.8. Since 827 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, or 23, we know that 827 is a prime number.

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Most numbers greater than 2 can be written as the sum of consecutive numbers. How can you know what those consecutive numbers are? By factoring, of course! Let’s take 826 as an example.

To find consecutive numbers that add up to 826, we are only interested in its odd factors that are less than or equal to 40 AND its factor that is the greatest power of 2. (We arrived at the number 40 because the largest triangular number less than 826 is the 40th triangular number, 820. We could also find the number 40 if we round √(1 + 826×2) – 1 to the nearest whole number.)

The factor of 820 that is the greatest power of 2 is 2. When we double that greatest power of 2, we get 4. The odd factors of 826 that are less than or equal to 40 are 1 and 7. Now we don’t ever count a number being the sum of just 1 consecutive number. So for 826, we are interested in just three numbers, all of which are less than or equal to the maximum number allowable, 40. Those numbers are 7, 1×4, and 7×4.

Thus, we can conclude that 826 can be written as the sum of 4 consecutive numbers, 7 consecutive numbers, and 28 consecutive numbers. Can you figure out what all those consecutive numbers are? How are the consecutive number sums derived from an odd factor different from the sums derived from an even number?

I’ve written out the sum of 4 consecutive numbers as an example and given some hints to help you figure out or check your answer to those two questions:

  • 826÷4 = 206.5, and that number lies right smack in the middle of the 4 consecutive numbers that make this sum: 205 + 206 + 207 + 208 = 826. Note that 205 + 208 and 206 + 207 both add up to 413, a factor of 826.
  • 826÷7 = 118, which is the exact middle number of the 7 consecutive numbers that sum up to 826. Note that 7 × 118 = 826.
  • 826÷28 = 29.5 so the 14 numbers from 16 to 29 plus the 14 numbers from 30 to 43 make the 28 numbers from 16 to 43 that add up to 826. Note that 16 + 43 = 59, a factor of 826.

Here’s today’s puzzle:

Print the puzzles or type the solution on this excel file: 10-factors-822-828

In order for (821, 823, 827, 829) to be the 4th prime decade, 825 or one of the even numbers between 820 and 830 has to be divisible by 7.

826 is the one that answered that call. Here is the 7 divisibility trick applied to 826:

  • 82-2(6) = 70, a number divisible by 7, so 826 is divisible by 7.

Why does 826 become the palindrome 181 in BASE 25? Because 1(25²) + 8(25¹) + 1(25º) = 826

Let’s begin with 826’s factoring information:

  • 826 is a composite number.
  • Prime factorization: 826 = 2 × 7 × 59
  • The exponents in the prime factorization are 1, 1, and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 = 8. Therefore 826 has exactly 8 factors.
  • Factors of 826: 1, 2, 7, 14, 59, 118, 413, 826
  • Factor pairs: 826 = 1 × 826, 2 × 413, 7 × 118, or 14 × 59
  • 826 has no square factors that allow its square root to be simplified. √826 ≈ 28.7402

 

Numbers ending in 00, 25, 50, and 75 can be divided evenly by 25. How much is 825 divided by 25? That quotient is the same as the answer to “how many quarters are in $8.25?” (A quarter is ¼ of a dollar and is written .25 or 25¢.)

You probably could visualize the answer in your head even if I hadn’t included a picture! That’s why I often ask kids the how-many-quarters question when they are stumped dividing something by 25 . It seems that kids are always able to give the quotient after that question. Notice that “8.25 ÷ .25 =” and  “8 ¼ ÷ ¼ =” have the same answer, too. You can also ask that how-many-quarters question to find the answer when something is divided by .25 or ¼.

It would almost be as easy to divide $8.26 or $8.39 by 25. The quotient would be the same as the problem above but with some loose change becoming the remainder. Using money for division problems could even help kids better understand dividends, divisors, quotients, and remainders.

Here’s an example of a how-many-quarters type question that will help you divide by 75, .75 or ¾.

We can count and see that there are 11 sets of 3 quarters in $8.25. That means that $8.25 ÷ .75 is 11. It also means that 8¼ ÷ ¾ = 11.

Dividing by fractions can be a very abstract concept for students, and they may ask questions like, “What does 8¼ ÷ 1¼ even mean?” Again quarters come to the rescue! 5 quarters can be so much more friendly than 1¼ is. Shorthand for 5 quarters is the fraction, 5/4. Since they have the same denominators, dividing 8¼ by 1¼ is the same as dividing 33 by 5:

Kids think money is more fun than math, but money is just a subset of mathematics which is full of lots of other fun topics, too. Here are a couple of ways other educators have used money to teach a math topic.

Jen of Beyond Tradit’l Math shared her way to teach subtracting decimals using money. Her method will surely captivate any child who tries it and even make regrouping fun:

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Robert Kaplinsky uses several very short videos to keep students engaged without them actually touching any money. Check out the replies, too. Paula Beardell Krieg’s excellent $1.00 art project is there, and that would be fun for anyone 2nd grade or older to do:

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Now back to the number 825.

Clearly 825 has to be divisible by both 5 AND 3 in order for (821, 823, 827, 829) to be the fourth prime decade, which it is.

  • The last digit of 825 is 5, so it is divisible by 5.
  • 8 + 2 + 5 = 15, a multiple of 3, so 825 is divisible by 3.
  • 8 – 2 + 5 = 11, so 825 is divisible by 11.

All numbers ending in 00, 25, 50, or 75 can have their square roots simplified. If you were trying to simplify √825, you could visualize quarters in your mind to easily divide 825 by 25. Then √825 = (√25)(√33) = 5√33

Here is 825’s factoring information:

  • 825 is a composite number.
  • Prime factorization: 825 = 3 × 5 × 5 × 11, which can be written 825 = 3 × 5² × 11
  • The exponents in the prime factorization are 1, 2, and 1. Adding one to each and multiplying we get (1 + 1)(2 + 1)(1 + 1) = 2 × 3 × 2 = 12. Therefore 825 has exactly 12 factors.
  • Factors of 825: 1, 3, 5, 11, 15, 25, 33, 55, 75, 165, 275, 825
  • Factor pairs: 825 = 1 × 825, 3 × 275, 5 × 165, 11 × 75, 15 × 55, or 25 × 33
  • Taking the factor pair with the largest square number factor, we get √825 = (√25)(√33) = 5√33 ≈ 28.72281

 

 

824 and Level 3

824 is the sum of all the prime numbers from 61 all the way to 103, which just happens to be one of its prime factors!

Print the puzzles or type the solution on this excel file: 10-factors-822-828

824 is a leg in a few Pythagorean triples:

  • 618-824-1030 because that is 206 times (3-4-5)
  • 824-1545-1751 because that is 103 times (8-15-17)
  • 824-10593-10625 because 2(103)(4) = 824
  • 824-21210-21226 because 105² – 101² = 824
  • 824-42432-42440 because 2(206)(2) = 824
  • 824-84870-84874 because 207² – 205² = 824
  • Primitive 824-169743-169745 because 2(412)(1) = 824

Five of those triples were derived directly from 824’s factor pairs.

Two of the triples were derived indirectly:

  • What is (105+101)/2, (105-101)/2?
  • Also, what is (207+205)/2, (207+205)/2?

The answer to both questions is a factor pair of 824.

You can read more about finding Pythagorean triples for numbers that are divisible by 4 here.

  • 824 is a composite number.
  • Prime factorization: 824 = 2 × 2 × 2 × 103, which can be written 824 = 2³ × 103
  • The exponents in the prime factorization are 3 and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1) = 4 × 2 = 8. Therefore 824 has exactly 8 factors.
  • Factors of 824: 1, 2, 4, 8, 103, 206, 412, 824
  • Factor pairs: 824 = 1 × 824, 2 × 412, 4 × 206, or 8 × 103
  • Taking the factor pair with the largest square number factor, we get √824 = (√4)(√206) = 2√206 ≈ 28.7054

 

 

823 and Level 2

All of the odd numbers between 820 and 830, except 825, are prime numbers. That makes (821, 823, 827, 829) the fourth prime decade.

Print the puzzles or type the solution on this excel file: 10-factors-822-828

 

  • 823 is a prime number.
  • Prime factorization: 823 is prime.
  • The exponent of prime number 823 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 823 has exactly 2 factors.
  • Factors of 823: 1, 823
  • Factor pairs: 823 = 1 x 823
  • 823 has no square factors that allow its square root to be simplified. √823 ≈ 28.687977

How do we know that 823 is a prime number? If 823 were not a prime number, then it would be divisible by at least one prime number less than or equal to √823 ≈ 28.7. Since 823 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, or 23, we know that 823 is a prime number.

822 and Level 1

The sum of 822’s digits is 12, a number divisible by 3. That means that even number 822 can be evenly divided by 2, 3, and 6.

822 is the sum of the 12 prime numbers from 43 to 97.

822 is palindrome 212 in base 20 because 2(20²) + 1(20¹) + 2(20º) = 822.

Print the puzzles or type the solution on this excel file: 10-factors-822-828

  • 822 is a composite number.
  • Prime factorization: 822 = 2 x 3 x 137
  • The exponents in the prime factorization are 1, 1, and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 x 2 x 2 = 8. Therefore 822 has exactly 8 factors.
  • Factors of 822: 1, 2, 3, 6, 137, 274, 411, 822
  • Factor pairs: 822 = 1 x 822, 2 x 411, 3 x 274, or 6 x 137
  • 822 has no square factors that allow its square root to be simplified. √822 ≈ 28.67054

The most famous Pythagorean triple is 3-4-5.

Perhaps you know that of the three numbers in EVERY Pythagorean triple at least one of them will be divisible by 3, at least one of them will be divisible by 4, and at least one of them will be divisible by 5. That’s obvious for triple 3-4-5.

Here’s another example: Pythagorean triple 11-60-61. Two of those numbers are prime numbers, yet 60 is divisible by 3, 4, AND 5.

If the Pythagorean triple is a primitive, something else amazing happens:

That’s amazing all by itself, but that’s only part of the picture. Let’s look at the complete picture specifically using the triple with hypotenuse 821:

  • 25² + 14² = 821
  • 429² + 700² = 821²
  • 429700821 can be calculated from 25² – 14², 2(25)(14), 25² + 14²
  • 821 + 700 = 1521
  • √1521 = 39 which is 25 + 14
  • 429 ÷ 39 = 11

WHY does this happen to primitive Pythagorean triples?

It happened for 429700821 because 700 82125² + 14² + 2(25)(14) = (25 + 14)².

After all (a + b)² = a² +2ab + b² is always true.

And it can be rearranged: (a + b)² = a² + b² +2ab.

You can use a similar proof whenever the element of the Primitive triple that is divisible by 4 can be expressed as 2ab.

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But it is a different, and perhaps simpler, story for many triples such as 94041 which was calculated using (9)(1), (9² – 1²)/2, (9² + 1²)/2.

In that case 40 + 41(9² – 1²)/2 + (9² + 1²)/2 = (9² – 1² + 9² + 1²)/2 = 9².

You can use a similar proof whenever the element of the Primitive triple that is divisible by 4 can be expressed as (a² – b²)/2.

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Go ahead, take ANY primitive Pythagorean triple. Add the leg that is divisible by 4 to its hypotenuse. You will get a perfect square. Here’s a few more examples:

Similarly primitive triples that have been multiplied by a square number will also produce a perfect square, but you’ll have to be careful which leg you add to the hypotenuse if you multiplied by a square number that is a multiple of 4.

For example, 9(3-4-5) = 27-36-45 and 36 + 45 = 81, a square number.

But 4(3-4-5) = 12-16-20 so each number is divisible by 4. Note that 16 + 20 = 36, a square number, but 12 + 20 does not.

Here’s a few other essential facts about the number 821:

All of the odd numbers between 820 and 830, except 825, are prime numbers. Thus 821, 823, 827, 829 is the fourth prime decade.

  • 821 is a prime number.
  • Prime factorization: 821 is prime.
  • The exponent of prime number 821 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 821 has exactly 2 factors.
  • Factors of 821: 1, 821
  • Factor pairs: 821 = 1 x 821
  • 821 has no square factors that allow its square root to be simplified. √821 ≈ 28.65309756.

How do we know that 821 is a prime number? If 821 were not a prime number, then it would be divisible by at least one prime number less than or equal to √821 ≈ 28.7. Since 821 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, or 23, we know that 821 is a prime number.

Here’s another way we know that 821 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 25² + 14² = 821 with 25 and 14 having no common prime factors, 821 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √821 ≈ 28.7. Since 821 is not divisible by 5, 13, or 17, we know that 821 is a prime number.

 

 

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