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

—————————————————————————————————

• 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

—————————————————————————————————

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.

 

 

 

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

—————————————————————————————————

• 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

—————————————————————————————————

76 can be evenly divided by 4 so 476 is also divisible by 4. Since 476 has a square factor, √476 can be simplified. Reducing square roots is easy if you make a cake: First divide 476 by its smallest square factor, 4, to get 119, a number that doesn’t have any square factors. Dividing 119 by its smallest prime factor is optional, but can help you see that 119, indeed, has no square factors. Last of all, take the square root of everything on the outside of the cake and multiply them together. √476 = (√4)(√(7 x 17) = (√4)(√119) = 2√119.

Here is today’s factoring puzzle:

Print the puzzles or type the solution on this excel file: 10 Factors 2015-04-27

—————————————————————————————————

• 476 is a composite number.
• Prime factorization: 476 = 2 x 2 x 7 x 17, which can be written 476 = (2^2) x 7 x 17
• 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 476 has exactly 12 factors.
• Factors of 476: 1, 2, 4, 7, 14, 17, 28, 34, 68, 119, 238, 476
• Factor pairs: 476 = 1 x 476, 2 x 238, 4 x 119, 7 x 68, 14 x 34, or 17 x 28
• Taking the factor pair with the largest square number factor, we get √476 = (√4)(√119) = 2√119 ≈ 21.81742

—————————————————————————————————

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.

468 is the sum of some consecutive primes. One of my readers posted what those primes are in the comments.

468 is 3333 in base 5. (Thank you OEIS.org for that cool fact.) Here’s proof going from right to left using some easy division problems. (See 3333 at the bottom.)

And here’s proof going from left to right using some more difficult division problems. (See 3333 at the top.)

—————————————————————————————————

The last two digits of 468 can be evenly divided by 4, so 468 is also divisible by 4.

468 is made from three consecutive even numbers so it is divisible by 3. Since the middle digit of the three consecutive even numbers is divisible by 3, we know that 468 can also be evenly divided by 9.

Let’s use those two facts to simplify the square root of 468 using the modified cake method. If you’re very confident in your ability to divide, you can make a one-layer cake and simply divide 468 by 36 to get 13. Then take the square root of everything on the outside of the cake and multiply them together: √468 = (√36)(√13) = 6√13

Many people will feel more comfortable making a two layer cake by dividing first by 4 and then by 9 as illustrated below:

Then to simplify √468, take the square root of everything on the outside of the cake and multiply those square roots together: √468 = (√4)(√9)(√13) = (2 x 3)(√13) = 6√13

You only need to know multiplication facts up to 12 x 12 to solve this factoring puzzle:

Print the puzzles or type the solution on this excel file:  12 Factors 2015-04-20

—————————————————————————————————

• 468 is a composite number.
• Prime factorization: 468 = 2 x 2 x 3 x 3 x 13, which can be written 468 = (2^2) x (3^2) x 13
• The exponents in the prime factorization are 2, 2 and 1. Adding one to each and multiplying we get (2 + 1)(2 + 1)(1 + 1) = 3 x 3 x 2 = 18. Therefore 468 has exactly 18 factors.
• Factors of 468: 1, 2, 3, 4, 6, 9, 12, 13, 18, 26, 36, 39, 52, 78, 117, 156, 234, 468
• Factor pairs: 468 = 1 x 468, 2 x 234, 3 x 156, 4 x 117, 6 x 78, 9 x 52, 12 x 39, 13 x 36 or 18 x 26
• Taking the factor pair with the largest square number factor, we get √468 = (√36)(√13) = 6√13 ≈ 21.6333

—————————————————————————————————

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.

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

——————————————————————————————————————-

• 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

——————————————————————————————————————-

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.

Today my blog has had 42 referral views from twitter, mostly thanks to other people’s tweets. Thank you for the tweets and retweets. Also, welcome everyone who visits my blog because of them or for any other reason!

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

Let’s try some divisibility tricks on the number 429:

4 + 2 is 6, a multiple of three, so 429 is a divisible by three. Why didn’t I include 9 in the sum? Because I already know that 9 is a multiple of three, so it is redundant to include it!

Now try another divisibility trick on 429: I’ve put the odd numbered digits in bold and the even numbered digits in regular type. If we sum the odd numbered digits and subtract the even numbered digit, we get 4 + 9 – 2 = 11, a multiple of 11. That means that 429 is also divisible by 11!

• 429 is a composite number.
• Prime factorization: 429 = 3 x 11 x 13
• 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 429 has exactly 8 factors.
• Factors of 429: 1, 3, 11, 13, 33, 39, 143, 429
• Factor pairs: 429 = 1 x 429, 3 x 143, 11 x 39, or 13 x 33
• 429 has no square factors that allow its square root to be simplified. √429 ≈ 20.7123

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.