453 Happy Golden Anniversary to Alan and Jill

/ ivasallay / 2 Comments

Today Alan and Jill Parr celebrate 50 years of marriage. I wanted to make a puzzle for their anniversary so I asked Alan for their favorite colors. He replied, “Colours? Nothing in particular, but I suppose I’d go for orange, and Jill would opt for something delicate and tasteful.”

Orange is much bolder than it is delicate so I went with his suggestion of “nothing in particular”: It’s their golden anniversary. Gold and blue go well together.

Golden Wedding Anniversary Puzzle

Since most people don’t want to solve puzzles on their anniversaries, I will include these puzzles in an excel file next week and wait until April 17th to update this post with the solutions.

Instead I’ve made an orange graphic with an interesting fact about the number 453 that I read at OEIS.org.

453 n, 2n, 6n

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  • 453 is a composite number.
  • Prime factorization: 453 = 3 x 151
  • The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 x 2 = 4. Therefore 453 has exactly 4 factors.
  • Factors of 453: 1, 3, 151, 453
  • Factor pairs: 453 = 1 x 453 or 3 x 151
  • 453 has no square factors that allow its square root to be simplified. √453 ≈ 21.2838

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Golden Wedding Anniversary Factors

452 and Level 4

/ ivasallay / 2 Comments

52 is divisible by 4 because 2 is an even number that is not divisible by 4, and the digit before it, 5, is an odd number. Thus 452 and ANY OTHER number ending in 52 is divisible by 4.

Here is another divisibility trick: 1% of all numbers end in the digits 52, which is NOT divisible by 8.  No matter how long the number is, if the digit immediately preceding those ending digits of 52 is odd, then that number will be divisible by 8, and if that digit is even, the number will NOT be divisible by 8. Thus 452 is NOT divisible by 8.

Live, Love, Laugh recently wrote a post about the Find the Factors blog. Check them out.

This Level 4 puzzle is a little tougher than usual, but if you’ve done other Level 4 puzzles, I think you can still handle it.

452 Puzzle

Print the puzzles or type the factors on this excel file:  12 Factors 2015-04-06

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  • 452 is a composite number.
  • Prime factorization: 452 = 2 x 2 x 113, which can be written 452 = (2^2) x 113
  • The exponents in the prime factorization are 2 and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1) = 3 x 2  = 6. Therefore 452 has exactly 6 factors.
  • Factors of 452: 1, 2, 4, 113, 226, 452
  • Factor pairs: 452 = 1 x 452, 2 x 226, or 4 x 113
  • Taking the factor pair with the largest square number factor, we get √452 = (√4)(√113) = 2√113 ≈ 21.2603

Since 452 has only 6 factors, we would get a two-layer cake when we use the cake method to find its factors. Dividing 452 by 4 is easier than dividing it by 2 twice, so I’ve modified the cake method to get just a one-layer cake to find the square root of 452. Since 113 is a prime number, no other divisions are possible.

452 square root

To simplify the square root of 452, simply take the square root of every number on the outside of the cake. Thus √452 = (√4)(√113) = 2√113.

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452 Logic

451 and Level 3

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Many fractions repeat a sequence of numbers forever when they are expressed as decimals:

  • 1/3 in decimal form has an unending series of 3’s after the decimal point
  • 1/11 is .090909… repeated forever. We say it has period two because exactly two digits repeat forever.
  • 1/7 is .142857142857… repeated forever. It has period six because those six unique digits repeat forever.
  • 1/451 has TEN digits repeat forever and ever. 451 is the smallest denominator whose decimal has ten digits repeating.

451 denominator

451 Puzzle

Print the puzzles or type the factors on this excel file:  12 Factors 2015-04-06

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  • 451 is a composite number.
  • Prime factorization: 451 = 11 x 41
  • The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 x 2 = 4. Therefore 451 has exactly 4 factors.
  • Factors of 451: 1, 11, 41, 451
  • Factor pairs: 451 = 1 x 451 or 11 x 41
  • 451 has no square factors that allow its square root to be simplified. √451 ≈ 21.2368

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A Logical Approach to solve a FIND THE FACTORS puzzle: Find the column or row with two clues and find their common factor. Write the corresponding factors in the factor column (1st column) and factor row (top row).  Because this is a level three puzzle, you have now written a factor at the top of the factor column. Continue to work from the top of the factor column to the bottom, finding factors and filling in the factor column and the factor row one cell at a time as you go.

451 Factors

How to Simplify √450

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What’s easier – dividing a number by 2 twice or dividing by 4 once? Most people would agree that dividing by a single digit number like 9 one time is easier than dividing by 3 two times. It also cuts the chance of making a mistake in half.

To simplify square roots, I’ve modified the cake method to look for some specific SQUARE factors rather than beginning with ALL of its prime factors.

Divisibility tricks let me know which square factors to try. I always divide out easy and very common perfect squares 100, 4, 9, and 25 first. (About 82% of reducible square roots are divisible by 4 and/or 9.) Once any of those that apply have been divided out, I look to see if 6 or 10 can be divided out because its easier to divide by 6 or 10 once than to divide by 2 and then by 3 or 5. Only after I have divided out those very easy divisors will I look to divide out any remaining prime factors 2, 3, 5, 7, 11, and so forth. Let me demonstrate this method to simplify √450.

  • 450 doesn’t end in 00, so it’s not divisible by 100.
  • The last two digits, 50, are not divisible by 4, so 450 is not divisible by 4.
  • 4 + 5 + 0 = 9, so 450 is divisible by 9. Therefore, I do a simple division problem, 450 ÷ 9 = 50, leaving room on the page to do any other needed division problems above it.
  • The previous quotient, 50, is not divisible by 9, but any number ending in 25, 50, or 75 is divisible by 25, so I divide it by 25 and get 2. Now I am finished dividing.
  • The numbers on the outside of the cake are 9, 25, and 2. I take the square root of each of those numbers and get 3, 5, and √2. The product of those square roots is 15√2. Thus √450 = 15√2.

450 Cake

Most people have been taught to use a factor tree to find square roots. This is probably 450’s most common factor tree:

450 Factor Tree

I like how much more compact and clear this modified cake method is instead. It may take some practice to get used to it, but I will show more examples of it in the future to make it more familiar.

The following puzzle doesn’t have anything to do with the number 450 except it’s the number I gave it to distinguish it from every other puzzle I make. I put the factors of 450 immediately after the puzzle to separate the puzzle from its solution that I add to the post the next day.

450 Puzzle

Print the puzzles or type the factors on this excel file:  12 Factors 2015-04-06

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  • 450 is a composite number.
  • Prime factorization: 450 = 2 x 3 x 3 x 5 x 5, which can be written 450 = 2 x (3^2) x (5^2)
  • The exponents in the prime factorization are 1, 2 and 2. Adding one to each and multiplying we get (1 + 1)(2 + 1)(2 + 1) = 2 x 3 x 3 = 18. Therefore 450 has exactly 18 factors.
  • Factors of 450: 1, 2, 3, 5, 6, 9, 10, 15, 18, 25, 30, 45, 50, 75, 90, 150, 225, 450
  • Factor pairs: 450 = 1 x 450, 2 x 225, 3 x 150, 5 x 90, 6 x 75, 9 x 50, 10 x 45, 15 x 30 or 18 x 25
  • Taking the factor pair with the largest square number factor, we get √450 = (√225)(√2) = 15√2 ≈ 21.2132

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450 Factors

449 and Level 1

/ ivasallay / 2 Comments

449 is the sum of consecutive prime numbers. How many and what are they? Check the comments to see the answer.

449 Puzzle

Print the puzzles or type the factors on this excel file:  12 Factors 2015-04-06

449 = 20² + 7² so 449 is the hypotenuse of a Pythagorean triple. That triple will be 2(20)(7), 20² – 7², 20² + 7² which is primitive triple 280-351-449.

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  • 449 is a prime number.
  • Prime factorization: 449 is prime.
  • The exponent of prime number 449 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 449 has exactly 2 factors.
  • Factors of 449: 1, 449
  • Factor pairs: 449 = 1 x 449
  • 449 has no square factors that allow its square root to be simplified. √449 ≈ 21.1896

How do we know that 449 is a prime number? If 449 were not a prime number, then it would be divisible by at least one prime number less than or equal to √449 ≈ 21.1896. Since 449 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, or 19, we know that 449 is a prime number.

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449 Factors

448 Because He Lives

/ ivasallay / 1 Comment

Happy Easter! The song, “Because He Lives,” is especially appropriate for Easter.

The composer, Sally DeFord, has written some absolutely beautiful music. You can download PDF’s of ANY of her sheet music for FREE on her website. This particular arrangement is written for a duet, but there is also a version for SATB. There are PDF’s of this particular song translated into Spanish, Portuguese, and Finnish. One of the people who commented even gave a translation into Chinese.

Since this is my 448th post, I’ll give some information about the number 448.

448 is another Harshad number because 4 + 4 + 8 = 16, and 448 can be evenly divided by 16.

448 factor tree

  • 448 is a composite number.
  • Prime factorization: 448 = 2 x 2 x 2 x 2 x 2 x 2 x 7, which can be written 448 = (2^6) x 7
  • The exponents in the prime factorization are 6, and 1. Adding one to each and multiplying we get (6 + 1)(1 + 1) = 7 x 2 = 14. Therefore 448 has exactly 14 factors.
  • Factors of 448: 1, 2, 4, 7, 8, 14, 16, 28, 32, 56, 64, 112, 224, 448
  • Factor pairs: 448 = 1 x 448, 2 x 224, 4 x 112, 7 x 64, 8 x 56, 14 x 32, or 16 x 28
  • Taking the factor pair with the largest square number factor, we get √448 = (√64)(√7) = 8√7 ≈ 21.16601

447 and Level 6

/ ivasallay / 4 Comments

447 is the hypotenuse for one Pythagorean triple: 153-420-447. Each of the numbers in the triple are divisible by 3. If you divide each of them by 3, you will discover the Primitive triple that this Pythagorean triple is based on and find one of 447’s factor pairs.

447 Puzzle

Print the puzzles or type the factors on this excel file: 10 Factors 2015-03-30

  • 447 is a composite number.
  • Prime factorization: 447 = 3 x 149
  • The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 x 2 = 4. Therefore 447 has exactly 4 factors.
  • Factors of 447: 1, 3, 149, 447
  • Factor pairs: 447 = 1 x 447 or 3 x 149
  • 447 has no square factors that allow its square root to be simplified. √447 ≈ 21.1424

447 Logic

Eight Ways to Make 446 with Three Squares

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When I read that 446 is the smallest number that can be expressed as the sum of three squares eight different ways, I just had to find those eight ways for myself. I found them, and it was fun. I made a graphic showing the eight ways:

8 ways to make 446 with 3 squares

446 Puzzle

Print the puzzles or type the factors on this excel file: 10 Factors 2015-03-30

  • 446 is a composite number.
  • Prime factorization: 446 = 2 x 223
  • The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 x 2 = 4. Therefore 446 has exactly 4 factors.
  • Factors of 446: 1, 2, 223, 446
  • Factor pairs: 446 = 1 x 446 or 2 x 223
  • 446 has no square factors that allow its square root to be simplified. √446 ≈ 21.1187

446 Logic

 

 

445 and Level 4

/ ivasallay / 11 Comments

Supposedly there is something interesting about every number. What is interesting about the number 445? When you write it in base 9, the digits are reversed. How do you convert 445 from base 10 to base 9?

Solution: The powers of 9 less than 445 in descending order are 81, 9, and 1. We first divide 445 by 81, next we divide the remainder by 9, and lastly, we divide that remainder by 1 as illustrated below.

445 base 9

Thus 445 (base 10) = 544 (base 9).

445 Puzzle

Print the puzzles or type the factors on this excel file: 10 Factors 2015-03-30

  • 445 is a composite number.
  • Prime factorization: 445 = 5 x 89
  • The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 x 2 = 4. Therefore 445 has exactly 4 factors.
  • Factors of 445: 1, 5, 89, 445
  • Factor pairs: 445 = 1 x 445 or 5 x 89
  • 445 has no square factors that allow its square root to be simplified. √445 ≈ 21.0950

445 is the hypotenuse of four Pythagorean triples:

  • [267-356-445] which is [3-4-5] times 89
  • [195-400-445] which is [39-80-89] times 5
  • Primitive [84-437-445] (Note: 84 = 2(2 x 21) and 437 = 21^2 – 2^2, while 445 = 21^2 + 2^2)
  • and Primitive [203-396-445] (Note: 396 = 2(18 x 11) and 203 = 18^2 – 11^2, while 445 = 18^2 + 11^2)

445 Logic

 

444 and Level 3

/ ivasallay / 2 Comments

What are the factors of 444? I’m not going to tell you to add 4 + 4 + 4 to see if it is divisible by 3 or 9, because it so obvious, there are three 4’s and three of anything is divisible by 3. The only way three of something could be divisible by 9 is if it were 333 or 666 or 999 so 444 is not divisible by 9.

But try adding the digits for a completely different reason: 4 + 4 + 4 = 12. Guess what, 444 can be evenly divided by 12. Many numbers can be divided evenly by the sum of their digits. In recreational mathematics, numbers with this property are called Harshad numbers. (Saying something is recreational mathematics is NOT an April Fool’s Day joke. Many people actually enjoy mathematics, including me.)

This also is no April Fool’s Day joke: I had a grandson born today at 1:47 pm; He weighed 8 lbs 14 oz.

This puzzle was made for your recreational enjoyment:

444 Puzzle

Print the puzzles or type the factors on this excel file: 10 Factors 2015-03-30

  • 444 is a composite number.
  • Prime factorization: 444 = 2 x 2 x 3 x 37, which can be written 444 = (2^2) x 3 x 37
  • The exponents in the prime factorization are 2, 1, and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1)(1 + 1) = 3 x 2 x 2 = 12. Therefore 444 has exactly 12 factors.
  • Factors of 444: 1, 2, 3, 4, 6, 12, 37, 74, 111, 148, 222, 444
  • Factor pairs: 444 = 1 x 444, 2 x 222, 3 x 148, 4 x 111, 6 x 74, or 12 x 37
  • Taking the factor pair with the largest square number factor, we get √444 = (√4)(√111) = 2√111 ≈ 21.0713

A Logical Approach to solve a FIND THE FACTORS puzzle: Find the column or row with two clues and find their common factor. Write the corresponding factors in the factor column (1st column) and factor row (top row).  Because this is a level three puzzle, you have now written a factor at the top of the factor column. Continue to work from the top of the factor column to the bottom, finding factors and filling in the factor column and the factor row one cell at a time as you go.

444 Factors