460 is the sum of consecutive prime numbers. Check the comments because one of my readers was able to find what those consecutive primes are.

Happy birthday to my son, Tim. I have two different cakes for you in this post. A cake puzzle and a simplified square root that uses the cake method that I’ve modified.

This puzzle will be included in an excel file of puzzles 12 Factors 2015-04-20.

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When we simplify square roots, we want to do as few divisions as possible. Since 60 can be evenly divided by perfect square 4, we know that 460 is also divisible by 4. Let’s use that fact to find its square root:

The quotient, 115, may be too large for us to know if it has any square factors. Since it isn’t divisible by 4, 9, or 25, let’s make a second layer to our cake as we divide it by its largest prime factor, 5.

Since the new quotient, 23, is a prime number, let’s revert back to the previous cake and take the square root of everything on the outside of the one layer cake: √460 = (√4)(√115) = 2√115.

• 460 is a composite number.
• Prime factorization: 460 = 2 x 2 x 5 x 23, which can be written 460 = (2^2) x 5 x 23
• 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 460 has exactly 12 factors.
• Factors of 460: 1, 2, 4, 5, 10, 20, 23, 46, 92, 115, 230, 460
• Factor pairs: 460 = 1 x 460, 2 x 230, 4 x 115, 5 x 92, 10 x 46, or 20 x 23
• Taking the factor pair with the largest square number factor, we get √460 = (√4)(√115) = 2√115 ≈ 21.4476

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Here’s the order the factors were found:

458 = (13^2) + (17^2). It is the hypotenuse of this Pythagorean triple: 120-442-458.

Print the puzzles or type the solution on this excel file:  10 Factors 2015-04-13

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• 458 is a composite number.
• Prime factorization: 458 = 2 x 229
• 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 458 has exactly 4 factors.
• Factors of 458: 1, 2, 229, 458
• Factor pairs: 458 = 1 x 458 or 2 x 229
• 458 has no square factors that allow its square root to be simplified. √458 ≈ 21.4009

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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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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.

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.

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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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.

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.