The exponent of prime number 463 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 463 has exactly 2 factors.
Factors of 463: 1, 463
Factor pairs: 463 = 1 x 463
463 has no square factors that allow its square root to be simplified. √463 ≈ 21.5174
How do we know that 463 is a prime number? If 463 were not a prime number, then it would be divisible by at least one prime number less than or equal to √463 ≈ 21.5174. Since 463 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, or 19, we know that 463 is a prime number.
462 is the sum of consecutive prime numbers two different ways. Check the comments to see what those ways are.
Divisibility tricks:
462 is even, so it is divisible by 2.
The sum of the odd numbered digits, 4 + 2 is 6, which is the 2nd digit, so 462 is divisible by 11.
Since both of those 6’s above are divisible by 3, then 462 is divisible by 3.
Separate the last digit from the rest and double it. 462 → 46 and 2; Doubling 2, gives us 4. Now subtract that 4 from the remaining digits: 46 – 4 = 42 which is divisible by 7, so 462 is divisible by 7.
Since 462 = 21 × 22, we know that it is two times the 21st triangular number, and it is the sum of the first 21 even numbers.
The exponents in the prime factorization are 1, 1, 1, and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1)(1 + 1) = 2 x 2 x 2 x 2 = 16. Therefore 462 has exactly 16 factors.
The exponent of prime number 461 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 461 has exactly 2 factors.
Factors of 461: 1, 461
Factor pairs: 461 = 1 x 461
461 has no square factors that allow its square root to be simplified. √461 ≈ 21.4709
How do we know that 461 is a prime number? If 461 were not a prime number, then it would be divisible by at least one prime number less than or equal to √461 ≈ 21.4709. Since 461 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, or 19, we know that 461 is a prime number.
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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.
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.
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.
459 is the hypotenuse of this Pythagorean triple: 216-405-459.
What is the greatest common factor of those three numbers?
The GCF has to be a factor of the smallest number, 216, and it has to be an odd number because at least one of the other numbers is odd. Let’s factor out the even factors of 216 to find its greatest odd factor:
216 can be evenly divided by 4 because the last two digits form a multiple of 4.
It can also be evenly divided by 8 because 16 is a multiple of 8 and the 3rd from the right digit is even.
216 ÷ 8 = 27.
Check to see if the other two numbers in the triple are divisible by 27, and you will see that 27 is the GCF of 216-405-459.
To solve this puzzle ask yourself:
What is a common factor of 6, 14, 10, and 16? What about 3, 18, 6, and 30? And what is a common factor of 9, 36, 45, 63, 54, 90, 72? In each case, the common factor has to be a factor of the smallest number on the list, and if any of the numbers on the list are odd, it has to be an odd number. (For level 1 and level 2 puzzles, that factor will oftrn be the greatest common factor of all the numbers in a particular row or column.)
459 cannot be evenly divided by 100 or by 4, but it is divisible by 9. To find it square root, let’s first divide 459 by 9:
The quotient, 51, is small enough that we can recognize that it cannot be evenly divided by any square number less than it. Thus we take the square root of everything on the outside of the cake and get √459 = (√9)(√51) = 3√51.
459 is a composite number.
Prime factorization: 459 = 3 x 3 x 3 x 17, which can be written 459 = (3^3) x 17
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 459 has exactly 8 factors.
Factors of 459: 1, 3, 9, 17, 27, 51, 153, 459
Factor pairs: 459 = 1 x 459, 3 x 153, 9 x 51, or 17 x 27
Taking the factor pair with the largest square number factor, we get √459 = (√9)(√51) = 3√51 ≈ 21.4243
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
457 = 4² + 21², and it is the hypotenuse of the primitive Pythagorean triple 168-425-457. Also, 457 is the sum of some consecutive prime numbers. One of my readers posted those primes in the comments.
A long time ago I decided that Pythagorean triples could make a great logic puzzle, so I created one. You can see it directly underneath the following directions:
This puzzle is NOT drawn to scale. Angles that are marked as right angles are 90 degrees, but any angle that looks like a 45 degree angle, isn’t 45 degrees. Lines that look parallel are NOT parallel. Shorter looking line segments may actually be longer than longer looking line segments. Most rules of geometry do not apply here: in fact non-adjacent triangles in the drawing might actually overlap.
No geometry is needed to solve this puzzle. All that is needed is the table of Pythagorean triples under the puzzle. The puzzle only uses triples in which each leg and each hypotenuse is less than 100 units long. The puzzle has only one solution.
If any of these directions are not clear, let me know in the comments. I will NOT be publishing the solution to this puzzle, but I will allow anyone who desires to put any or all of the missing values in the comments. Also, the comments will help me determine if I should publish another puzzle like this one.
The exponent of prime number 457 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 457 has exactly 2 factors.
Factors of 457: 1, 457
Factor pairs: 457 = 1 x 457
457 has no square factors that allow its square root to be simplified. √457 ≈ 21.3776
How do we know that 457 is a prime number? If 457 were not a prime number, then it would be divisible by at least one prime number less than or equal to √457 ≈ 21.3776. Since 457 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, or 19, we know that 457 is a prime number.
456 is the sum of consecutive prime numbers in two different ways. One of my readers listed those ways in the comments. The factors of 456 are at the end of the post.
Inchworm, inchworm,
Measuring the marigolds
You and your arithmetic will probably go far.
Two plus two is four
Four plus four is eight
Eight and eight is sixteen
Sixteen and sixteen is thirty-two.
Inchworm, inchworm,
Measuring the marigolds
Seems to me you’d stop and see
How beautiful they are.
Today I taught a class of three year olds about being thankful for birds, insects, and creeping things. To keep their attention, I used a variety of stories, riddles, books, and games. I also sang a few songs including this one about an inchworm who is very good at arithmetic. I think preschool children can still enjoy songs like this even if they don’t understand everything the song is about or even if they are wiggling as much as an inchworm while they listen to it. Here is the song sung by Danny Kaye from the movie Hans Christian Andersen:
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Now for the number 456. The last two digits can be evenly divided by four, so the entire number is divisible by four. Also since it is formed from three consecutive numbers, it is divisible by 3. However since the number in the middle of those consecutive numbers is not 3, 6, 9 or another multiple of 3, we know that 456 is NOT divisible by 9.
Because it is divisible by four, we will use that fact first to determine how to reduce its square root.
456 ÷ 4 = 114. Notice that 114 is even, but 14 can’t be evenly divided by 4, so 114 cannot be either. Also notice that 114 is still divisible by 3. If we’re not sure whether or not 114 has any square factors, we are less likely to make a mistake if we divide it by 6 once, instead of by 2 and then by 3.
114 ÷ 6 = 19, a prime number, and we are certain there were no other square factors. Since we know 19 x 6 = 114, let’s backtrack a little and go back to that original one layer cake:
Take the square root of everything on the outside of the cake and get √456 = (√4)(√114) = 2√114
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456 is a composite number.
Prime factorization: 456 = 2 x 2 x 2 x 3 x 19, which can be written 456 = (2^3) x 3 x 19
The exponents in the prime factorization are 3, 1, and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1)(1 + 1) = 4 x 2 x 2 = 16. Therefore 456 has exactly 16 factors.
I don’t mean to sound greedy, but if there were 13 days of Christmas instead of only 12, my true love would give me 455 gifts instead of only 364. That’s because the sum of the first 13 triangular numbers is 455. Come on, that’s 91 more gifts. Funny thing, 91 is one of the factors of 455. Also, I know I’m not the first person to notice that (13 x 14 x 15)/6 = 455. As I’m sure you can see, 455 is a fabulous tetrahedral number.
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 445 has exactly 8 factors.
Factors of 455: 1, 5, 7, 13, 35, 65, 91, 455
Factor pairs: 455 = 1 x 455, 5 x 91, 7 x 65, or 13 x 35
455 has no square factors that allow its square root to be simplified. √455 ≈ 21.3307
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!
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
