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 282 has 8 factors.

Factors of 282: 1, 2, 3, 6, 47, 94, 141, 282

Factor pairs: 282 = 1 x 282, 2 x 141, 3 x 94, or 6 x 47

282 has no square factors that allow its square root to be simplified. √282 ≈ 16.793

Today is Halloween. Give this Level 5 puzzle a try. . . Trick or Treat?

Prime factorization: 281 is prime and cannot be factored.

The exponent of prime number 281 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 281 has exactly 2 factors.

Factors of 281: 1, 281

Factor pairs: 281 = 1 x 281

281 has no square factors that allow its square root to be simplified. √281 ≈ 16.763

How do we know that 281 is a prime number? If 281 were not a prime number, then it would be divisible by at least one prime number less than or equal to √281 ≈ 16.763. Since 281 cannot be divided evenly by 2, 3, 5, 7, 11, or 13, we know that 281 is a prime number.

So far I have posted about one set of four and two sets of five consecutive reducible square roots. Sets of three consecutive reducible square roots are fairly common so I’ve ignored most of them. These consecutive square roots couldn’t be ignored:

The final square root features today’s prime number 281. Here are the prime factorizations and number of factors of each of these numbers:

Six is a popular number when counting the number of factors.

Prime factorization: 280 = 2 x 2 x 2 x 5 x 7, which can be written 280 = 2³ x 5 x 7

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 280 has 16 factors.

Factor pairs: 280 = 1 x 280, 2 x 140, 4 x 70, 5 x 56, 7 x 40, 8 x 35, 10 x 28, or 14 x 20

Taking the factor pair with the largest square number factor, we get √280 = (√4)(√70) = 2√70 ≈ 16.733

When I was a child, my brother showed me how he could multiply using his fingers. The same method is clearly illustrated here. If you haven’t seen it before, take a look. It isn’t the fastest method for multiplying, but it is still fun to see that it works. You can also practice the multiplication facts by finding the factors that solve this puzzle and then completing the multiplication table.

Prime factorization: 279 = 3 x 3 x 31, which can be written (3^2) x 31

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 279 has 6 factors.

Factors of 279: 1, 3, 9, 31, 93, 279

Factor pairs: 279 = 1 x 279, 3 x 93, or 9 x 31

Taking the factor pair with the largest square number factor, we get √279 = (√9)(√31) = 3√31 ≈ 16.703

Superman and his secret identity, Clark Kent

Superman has a secret identity, but did you know that 1, 2, 3, 4, and 5 also have secret identities? Before today, hardly anybody has known what their secret identities are, but I will reveal them to you now!

Let’s start with five. Its secret identity is zero. Five can change into a zero by smoothing its top and curving it down. When you count by fives you can see it changing into zero and back into five again as fast as Superman can change into Clark Kent. 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100, 105, and so forth.

By using its secret identity , we can easily add or subtract five from a number ending in five or zero! 5 + 5 = 10, 10 + 5 = 15, and 15 – 5 = 10; 10 – 5 = 5.

Have you ever noticed how easy it is to turn a 4 into a 9?

That was as easy as Superman becoming Clark Kent!

When you count by 5’s starting at 4, these numbers turn into each other over and over again: 4, 9, 14, 19, 24, 29, and so forth. When you add 5 to either 4 or 9, it will turn into the other: 4 + 5 = 9, 9 + 5 = 14. The same thing happens when you subtract five: 14 – 5 = 9; 9 – 5 = 4. There isn’t any reason to count up or down; just remember how easy it is to turn a 4 into a 9 and vice versa.

A 3 can easily turn into an 8:

Counting by 5’s starting at 3 we get: 3, 8, 13, 18, 23, 28, and so forth.

If you cut off the tail at the bottom of a two, it can easily turn into its secret identity, seven:

Counting by 5’s starting at 2, we get: 2, 7, 12, 17, 22, 27, and so forth.

2 + 5 = 7; 7 + 5 = 12 and 12 – 5 = 7; 7 – 5 = 2.

Here’s how to discover the secret identity for the number 1. Take a strip of paper that looks like the number 1 and follow these SAFE directions for curling ribbon (or paper) with scissors. Curl the bottom half of the number 1 as well as the top fourth of that number 1 with scissors to make that number 1 look just like the number 6.

Counting by 5’s starting at 1, we get: 1, 6, 11, 16, 21, 26, and so forth.

1 + 5 = 6; 6 + 5 = 11 and 11 – 5 = 6; 6 – 5 = 1.

When Superman puts on glasses and a suit, we see his secret identity. Did you notice that odd digits have even secret identities and even digits have odd secret identities? Once a child memorizes these digits and their secret identities, he or she will be SUPER at adding or subtracting 5 and will never need to count up or down to get the answer.

Using these number transformations can help children memorize other addition facts. Once they know how to Add 1, 2, 3 and 4 and how to add 5, they can use that information to add 6, 7, 8, 9, or 10 by breaking up those numbers into a smaller number plus five. For example:

6 + 7 = 6 + (2 + 5) = (6 + 2) + 5 = 8 + 5 = 13.

7 + 9 = 7 + (4 + 5) = (7 + 4) + 5 = 11 + 5 = 16.

Learning facts for 6, 7, 8, 9, and 10 this way isn’t any more complicated than the common core approach[6 + 7 = 6 + (6 + 1) = (6 + 6) + 1 = 12 + 1 = 13], but is perhaps not as easy as just memorizing those facts and using flash cards or other means to drill them permanently into the brain. Secret identities and “Flash” cards can be super fun.

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 278 has 4 factors.

Factors of 278: 1, 2, 139, 278

Factor pairs: 278 = 1 x 278 or 2 x 139

278 has no square factors that allow its square root to be simplified. √278 ≈ 16.673

A Logical Approach to FIND THE FACTORS: 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.

Prime factorization: 277 is prime and cannot be factored.

The exponent of prime number 277 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 277 has exactly 2 factors.

Factors of 277: 1, 277

Factor pairs: 277 = 1 x 277

277 has no square factors that allow its square root to be simplified. √277 ≈ 16.643

How do we know that 277 is a prime number? If 277 were not a prime number, then it would be divisible by at least one prime number less than or equal to √277 ≈ 16.643. Since 277 cannot be divided evenly by 2, 3, 5, 7, 11, or 13, we know that 277 is a prime number.

What can a level 2 puzzle do? Sometimes the “impossible” as this tweet suggests:

Prime factorization: 276 = 2 x 2 x 3 x 23, which can be written 276 = (2^2) x 3 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 276 has 12 factors.

Factor pairs: 276 = 1 x 276, 2 x 138, 3 x 92, 4 x 69, 6 x 46, or 12 x 23

Taking the factor pair with the largest square number factor, we get √276 = (√4)(√69) = 2√69 ≈ 16.613

I’m excited that resourceaholic included a FIND THE FACTORS puzzle in a gems post. Check it out! There are several other interesting resources there, too.

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