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.
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
Do you know who Superman is? I’ve asked many children in their first year of school that question. They know who he is, and they can tell me lots of things about him.
Superman’s secret identity is Clark Kent, but did you know that the numbers 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 even “Flash” cards can be super fun.
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.
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:
I recently read a post at mathfour.com that discussed the “basic” concept of odd and even numbers and children’s ability to understand the difference. The article made me very curious so I talked briefly to 45 first grade students about even and odd numbers. What did I find out?
Almost all of them had been introduced to the concept in kindergarten and knew that 1, 3, 5, 7, 9 are odd numbers while 2, 4, 6, 8, 10 are even.
A few accelerated learning students were able to explain to me that the one’s digit of a number determines if the number is even or odd,
But most of these first graders did not understand that fact because about a third of the students thought that 32 is odd!
One little girl explained to me how odd and even numbers alternate. She said, “If 99 is even, then 100 will be odd.” She remembered that concept but didn’t understand it well enough to apply it to the example she gave!
Even though odd and even numbers may be a difficult concept to learn, teach the concept and use it anyway. In fact, talk about it to preschoolers while you put on their socks, shoes, or mittens. One,_Two,_Buckle_My_Shoe.
Children learn to recite numbers in order before they learn how to count, and that helps them learn how to count and later how to add or subtract 1 from a number. I have tutored bewildered looking students who weren’t sure what to do with 8 + 1 = until I told them that 8 + 1 = means “what number comes right after 8 when you count?” Likewise, 8 – 1 = means “what number comes right before 8 when you count?” After hearing those questions, these students immediately knew the answer, and they didn’t count to find it.
Children who can quickly recite the odd numbers to 11 and the even numbers to 10 will have an easier time adding or subtracting two from a number. When they see 3 + 2 =, they can remember that 3 is odd and then ask themselves what odd number comes after 3. Likewise when they see 8 – 2, they can remember that 8 is even and recall that 6 is the even number right before 8.
The way I remember it, I was in second grade when I first was told that an even number plus an even number is even, an odd number plus an odd number is even, while an even number plus an odd number is odd. Any student learning to add or subtract would benefit from that tip.
Adding 3 to an odd number gives an even number, in fact, it’s the second even number after the original number. Adding 3 to an even number gives an odd number which is the second odd number after the original number. Subtracting 3 has the same rule, but substitute the word “before” for the word “after.”
Adding 4 to an odd number gives the second odd number after it while adding 4 to an even number gives the second even number after it. Subtracting 4 has a similar rule.
Adding 3 or 4 will mean additionally memorizing that 12 and 14 are even and 13 is odd, but that will be all a first grader needs to know about odd and even numbers. Later these two categories of numbers will be useful throughout their lives for many, many reasons.
What are some ways to help children to memorize odd and even numbers? Paula Beardell Krieg has designed the most captivating number line in the world.
The transformation can be done by a child or an adult. This number line that is made with envelopes is pretty enough to hang on a classroom wall, but it can fold up like a book, or be played with and changed so that real learning can take place. Paula Beardell Krieg shows several uses of it in her post, the-flux-capacity-of-an-artful-number-line, and promises to give directions on how to make one soon!
Try these rhymes: 0, 2, 4, 6, 8; Being EVEN is just great! 1, 3, 5, 7, 9; Being ODD is just fine!
Smartfirstgraders.com has several activities and rhymes to help students memorize the odd and even numbers.
Finally, if you clap when you say ODD, you will clap one time. 1 is an odd number.
If you clap when you say EVEN, you will clap two times, 2 is even.
And as mathfour.com pointed out in more detail then I’m showing here: ODD has 3 letters, and 3 is odd.
Also EVEN has 4 letters to help us remember that 4 is even.
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 274 has 4 factors.
Factors of 274: 1, 2, 137, 274
Factor pairs: 274 = 1 x 274 or 2 x 137
274 has no square factors that allow its square root to be simplified. √274 ≈ 16.553.
I recommend using logic and not guess and check to solve the Find the Factors puzzles, but the logic for the one below can be a little tricky. If necessary, scroll down for one piece of reasoning that may be helpful if you get stuck in the middle of trying to solve this one.