There is a relationship between the digits in 454 and the digits of its two prime factors, specifically 4 + 5 + 4 = 13 and 2 + 2 + 2 + 7 = 13. That means 454 is another Smith number.

I am excited that Jo Morgan of Resourceaholic has awarded this blog the 2015 Resource Gem Award for Bright Ideas!

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

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• 454 is a composite number.
• Prime factorization: 454 = 2 x 227
• 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 454 has exactly 4 factors.
• Factors of 454: 1, 2, 227, 454
• Factor pairs: 454 = 1 x 454 or 2 x 227
• 454 has no square factors that allow its square root to be simplified. √454 ≈ 21.3073

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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.

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.

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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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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:

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

431 is the sum of seven consecutive prime numbers beginning with prime number 47. What other prime numbers are in that sum? If you would like, you can type your answer in the comments.

2^431 (mod 431) = 2 so 431 is VERY likely a prime number. Scroll down past the puzzle to know for sure.

Print the puzzles or type the factors on this excel file:10 Factors 2015-03-16

• 431 is a prime number.
• Prime factorization: 431 is prime.
• The exponent of prime number 431 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 431 has exactly 2 factors.
• Factors of 431: 1, 431
• Factor pairs: 431 = 1 x 431
• 431 has no square factors that allow its square root to be simplified. √431 ≈ 20.7605

How do we know that 431 is a prime number? If 431 were not a prime number, then it would be divisible by at least one prime number less than or equal to √431 ≈ 20.7605. Since 431 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, or 19, we know that 431 is a prime number.

The last two digits of 424 can be evenly divided by 4 so 424 can also be evenly divided by 4. Also because 24 is divisible by 8 and because the digit preceding the 24 is even, 424 is also divisible by 8. All of the factors of 424 are listed below the puzzle.

Print the puzzles or type the factors on this excel file: 12 Factors 2015-03-09

• 424 is a composite number.
• Prime factorization: 424 = 2 x 2 x 2 x 53, which can be written 424 = (2^3) x 53
• The exponents in the prime factorization are 3 and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1) = 4 x 2 = 8. Therefore 424 has exactly 8 factors.
• Factors of 424: 1, 2, 4, 8, 53, 106, 212, 424
• Factor pairs: 424 = 1 x 424, 2 x 212, 4 x 106, or 8 x 53
• Taking the factor pair with the largest square number factor, we get √424 = (√4)(√106) = 2√106 ≈ 20.5913

399 is 400 – 1. Since 20 x 20 = 400, it is VERY easy to know one of the factor pairs for 399. It will be 399 = (20 – 1) x (20 + 1).

Print the puzzles or type the factors on this excel file: 10 Factors 2015-02-16

• 399 is a composite number.
• Prime factorization: 399 = 3 x 7 x 19
• 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 399 has exactly 8 factors.
• Factors of 399: 1, 3, 7, 19, 21, 57, 133, 399
• Factor pairs: 399 = 1 x 399, 3 x 133, 7 x 57, or 19 x 21
• 399 has no square factors that allow its square root to be simplified. √399 ≈ 19.975

2^391 (mod 391) = 179, not 2 so 391 is definitely a composite number. Scroll down below the puzzle to view its factors.

When I was in elementary school, my sister, Sue, taught me a song she learned in her junior high choir class. I’m not able to find these lyrics on the internet, but as best as I can remember, the words of the song were:

In the woods, in the spring, blooms the lovely wild rose. Every bud waking facing the sun. Hearts of spring, beat on thee, as the light of day grows. Dew drops glisten on you, Wild Rose. Fill the countryside with your lovely fragrance. Sun keeps your petals warm, leaves hide you from harm. Day will break, petals close, close them, oh, so tightly. Cover your beauty from the cool night. Birds, winging, singing through the woodland. Sleep and dream, sleep and dream, sleep and dream.

Print the puzzles or type the factors on this excel file:12 Factors 2015-02-09

• 391 is a composite number.
• Prime factorization: 391 = 17 x 23
• 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 391 has exactly 4 factors.
• Factors of 391: 1, 17, 23, 391
• Factor pairs: 391 = 1 x 391 or 17 x 23
• 391 has no square factors that allow its square root to be simplified. √391 ≈ 19.774

• Note that 17 + 3 = 20, and 20 + 3 = 23, and (20^2) – (3^2) = 400 – 9 = 391.
• 2(3*20), (20^2) – (3^2), (20^2) + (3^2) makes primitive Pythagorean triple 120-391-409.
• 391-76440-76441 is another primitive Pythagorean triple
• 184-345-391 is [8-15-17] multiplied by 23
• 391-3312-3335 is [17-144-145] multiplied by 23
• 391-4488-4505 is [23-264-265] multiplied by 17

391 is in this cool pattern: