490 is seventy times seven. A famous Bible verse mentioning this number is Matthew 18: 21 – 22.

This is the King James Version: “Then came Peter to him, and said, Lord, how oft shall my brother sin against me, and I forgive him? till seven times? Jesus saith unto him, I say not unto thee, Until seven times: but, Until seventy times seven.”

Most other Bible translation also have seventy times seven, but a few translations have only seventy seven. You can compare several different translations of Matthew 18: 22 here. Regardless of the translation, forgiveness is much more important than keeping count especially since all of us need to be forgiven as well.

To find √490 first notice that 490 can be evenly divided by 2, but not by 4. It also cannot be evenly divided by 9. It is divisible by 5, but since it doesn’t end in 25, 50, 75, or 00, it can only be divided by 5 once. Since it is divisible by 10, and dividing by 10 is so easy, I would make this one-layer cake:

Then I would take the square root of everything on the outside of the cake and multiply together to get √490 = (√10)(√49) = 7√10.

Print the puzzles or type the solution on this excel file: 10 Factors 2015-05-11

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• 490 is a composite number.
• Prime factorization: 490 = 2 x 5 x 7 x 7, which can be written 490 = 2 x 5 x (7^2)
• The exponents in the prime factorization are 1, 1, and 2. Adding one to each and multiplying we get (1 + 1)(1 + 1)(2 + 1) = 2 x 2 x 3 = 12. Therefore 490 has exactly 12 factors.
• Factors of 490: 1, 2, 5, 7, 10, 14, 35, 49, 70, 98, 245, 490
• Factor pairs: 490 = 1 x 490, 2 x 245, 5 x 98, 7 x 70, 10 x 49, or 14 x 35
• Taking the factor pair with the largest square number factor, we get √490 = (√49)(√10) = 7√10 ≈ 22.135943

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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.

 

 

 

489 is the ninth octahedral number.  Its square root starts out in quite a memorable way: √489 ≈ 22.113344, but after those six decimal places, it looks as irrational as can be.

 

Print the puzzles or type the solution on this excel file: 10 Factors 2015-05-11

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• 489 is a composite number.
• Prime factorization: 489 = 3 x 163
• 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 489 has exactly 4 factors.
• Factors of 489: 1, 3, 163, 489
• Factor pairs: 489 = 1 x 489 or 3 x 163
• 489 has no square factors that allow its square root to be simplified. √489 ≈ 22.113344

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Today’s Puzzle:

Print the puzzles or type the solution on this excel file: 12 Factors 2015-05-04

Here is a logical order to use the clues to find the solution to the puzzle:

How to Simplify the Square Root of 486:

To find √486, I like to first see if 486 can be evenly divided by square numbers 4 or 9 because about 82% of numbers that can have their square roots reduced are divisible by 4 and/or by 9. It is also very easy to tell if a number can be evenly divided by 4 or by 9.

86 is not divisible by 4, so 486 is not divisible by 4.

However, 4 + 8 + 6 = 18, a multiple of 9 so 486 is divisible by 9, and now we know for sure that its square root can be simplified. 486 ÷ 9 = 54, another multiple of 9 so we get this two-layer cake when we divide by 9 twice.

Now take the square root of everything on the outside of the cake, and we get √486 = (√9)(√9)(√6) = 9√6.

Factors of 486:

• 486 is a composite number.
• Prime factorization: 486 = 2 x 3 x 3 x 3 x 3 x 3, which can be written 486 = 2 x 3⁵.
• The exponents in the prime factorization are 1 and 5. Adding one to each and multiplying we get (1 + 1)(5 + 1) = 2 x 6 = 12. Therefore 486 has exactly 12 factors.
• Factors of 486: 1, 2, 3, 6, 9, 18, 27, 54, 81, 162, 243, 486
• Factor pairs: 486 = 1 x 486, 2 x 243, 3 x 162, 6 x 81, 9 x 54, or 18 x 27
• Taking the factor pair with the largest square number factor, we get √486 = (√81)(√6) = 9√6 ≈ 22.04540

Sum-Difference Puzzles:

6 has two factor pairs. One of those pairs adds up to 5, and the other one subtracts to 5. Put the factors in the appropriate boxes in the first puzzle.

486 has eight factor pairs. One of the factor pairs adds up to 45, and a different one subtracts to 45. If you can identify those factor pairs, then you can solve the second puzzle!

The second puzzle is really just the first puzzle in disguise. Why would I say that?

484 is the sum of 14 consecutive primes, and the first of those primes is one of its factors!

11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 = 484

Print the puzzles or type the solution on this excel file: 12 Factors 2015-05-04

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• 484 is a composite number.
• Prime factorization: 484 = 2 x 2 x 11 x 11, which can be written 484 = (2^2) x (11^2)
• The exponents in the prime factorization are 2 and 2. Adding one to each and multiplying we get (2 + 1)(2 + 1) = 3 x 3 = 9. Therefore 484 has exactly 9 factors.
• Factors of 484: 1, 2, 4, 11, 22, 44, 121, 242, 484
• Factor pairs: 484 = 1 x 484, 2 x 242, 4 x 121, 11 x 44, or 22 x 22
• 484 is a perfect square. √484 = 22

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4 + 8 = 12, a multiple of 3, so 483 can be evenly divided by 3. (It wasn’t necessary to add 4 + 8 + 3 because we already know 3 is divisible by 3.)

We can easily see that 483 is divisible by 7 if we apply the divisibility trick for 7. Separate the last number, 3, from the rest, double it, and subtract the double from the remaining numbers: 48 – (3 x 2) = 48 – 6 = 42, which is a multiple of 7. Thus 483 is divisible by 7.

Since 483 is 21 x 23, we can predict that 484 will be 22 x 22.

Print the puzzles or type the solution on this excel file: 12 Factors 2015-05-04

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• 483 is a composite number.
• Prime factorization: 483 = 3 x 7 x 23
• 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 483 has exactly 8 factors.
• Factors of 483: 1, 3, 7, 21, 23, 69, 161, 483
• Factor pairs: 483 = 1 x 483, 3 x 161, 7 x 69, or 21 x 23
• 483 has no square factors that allow its square root to be simplified. √483 ≈ 21.97726

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Can you find the greatest common factor of the three numbers in this Pythagorean triple 156-455-481? Or in Pythagorean triple 185-444-481?

481 is also the hypotenuse in two primitive Pythagorean triples: 319-360-481 and 31-480-481. Since they are primitive triples, their greatest common factor is 1.

Print the puzzles or type the solution on this excel file: 12 Factors 2015-05-04

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• 481 is a composite number.
• Prime factorization: 481 = 13 x 37
• 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 481 has exactly 4 factors.
• Factors of 481: 1, 13, 37, 481
• Factor pairs: 481 = 1 x 481 or 13 x 37
• 481 has no square factors that allow its square root to be simplified. √481 ≈ 21.93171

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