386 is even and composite.

Print the puzzles or type the factors on this excel file:12 Factors 2015-02-09

• 386 is a composite number.
• Prime factorization: 386 = 2 x 193
• 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 386 has exactly 4 factors.
• Factors of 386: 1, 2, 193, 386
• Factor pairs: 386 = 1 x 386 or 2 x 193
• 386 has no square factors that allow its square root to be simplified. √386 ≈ 19.647

Today’s Puzzle:

Print the puzzles or type the factors on this excel file: 10 Factors 2015-02-02

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

Factors of 384:

384 is even and, therefore, composite.

• 384 is a composite number.
• Prime factorization: 384 = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 3, which can be written 384 = (2^7) x 3
• The exponents in the prime factorization are 7 and 1. Adding one to each and multiplying we get (7 + 1)(1 + 1) = 8 x 2 = 16. Therefore 384 has exactly 16 factors.
• Factors of 384: 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 64, 96, 128, 192, 384
• Factor pairs: 384 = 1 x 384, 2 x 192, 3 x 128, 4 x 96, 6 x 64, 8 x 48, 12 x 32, or 16 x 24
• Taking the factor pair with the largest square number factor, we get √384 = (√6)(√64) = 8√6 ≈ 18.596

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.

384 has eight factor pairs. One of the factor pairs adds up to 40, and a different one subtracts to 40. 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?

More about the Number 384:

384 is in this cool pattern:

 

382 is another even composite number. Scroll past the puzzle for its factoring information.

Print the puzzles or type the factors on this excel file: 10 Factors 2015-02-02

• 382 is a composite number.
• Prime factorization: 382 = 2 x 191
• 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 382 has exactly 4 factors.
• Factors of 382: 1, 2, 191, 382
• Factor pairs: 382 = 1 x 382 or 2 x 191
• 382 has no square factors that allow its square root to be simplified. √382 ≈ 19.545

382 is in only one Pythagorean triple: 382 – 36480 – 36482 which is 2 times (191 – 18240 – 18241).

Is 381 Prime or composite? Its digits can be grouped into 3 and 81. Both groups are divisible by 3 so 381 is composite. Scroll down past the puzzle to view its factors.

Print the puzzles or type the factors on this excel file: 10 Factors 2015-02-02

• 381 is a composite number.
• Prime factorization: 381 = 3 x 127
• 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 381 has exactly 4 factors.
• Factors of 381: 1, 3, 127, 381
• Factor pairs: 381 = 1 x 381 or 3 x 127
• 381 has no square factors that allow its square root to be simplified. √381 ≈ 19.519

Pythagorean triples:

• 381 – 508 – 635 which is 127 times (3 – 4 – 5 )
• 381 – 24192 – 24195 which is 3 times (127 – 8064 – 8065)
• 381 – 8060 – 8069, a primitive
• 381 – 72580 – 72581, a primitive

380 is even so it has to be a composite number. Scroll down past the puzzle for all of its factors.

Print the puzzles or type the factors on this excel file: 10 Factors 2015-02-02

• 380 is a composite number.
• Prime factorization: 380 = 2 x 2 x 5 x 19, which can be written 380 = (2^2) x 5 x 19
• 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 380 has exactly 12 factors.
• Factors of 380: 1, 2, 4, 5, 10, 19, 20, 38, 76, 95, 190, 380
• Factor pairs: 380 = 1 x 380, 2 x 190, 4 x 95, 5 x 76, 10 x 38, or 19 x 20
• Taking the factor pair with the largest square number factor, we get √380 = (√4)(√95) = 2√95 ≈ 19.494

Since those consecutive numbers in the last factor pair are the last possibility on the chart, we know that 380 is the sum of the first 19 even numbers. Thus,

2 + 4 + 6 + 8 + 10 + 12 + 14 + 16 + 18 + 20 + 22 + 24 + 26 + 28 + 30 + 32 + 34 + 36 + 38 = 380. Those numbers can be rearranged this way:

2 + 36 + 4 + 34 + 6 + 32 + 8 + 30 + 10 + 28 + 12 + 26 + 14 + 24 + 16 + 22 + 18 + 20 + 38 = 380, which is more easily seen as the sum of ten 38’s.

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.

 

2^379 (mod 379) = 2 so 379 is VERY LIKELY a prime number. Scroll down below the puzzle to know for sure.

Alan Parr, who worked on some fun projects at NRICH, recently wrote about his game, Envelopes, on his blog. This easy-to-make game can be used to reinforce addition or multiplication facts, and it never looks anything like a worksheet.

Print the puzzles or type the factors on this excel file: 10 Factors 2015-02-02

• 379 is a prime number.
• Prime factorization: 379 is prime.
• The exponent of prime number 379 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 379 has exactly 2 factors.
• Factors of 379: 1, 379
• Factor pairs: 379 = 1 x 379
• 379 has no square factors that allow its square root to be simplified. √379 ≈ 19.468

How do we know that 379 is a prime number? If 379 were not a prime number, then it would be divisible by at least one prime number less than or equal to √379 ≈ 19.468. Since 379 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, or 19, we know that 379 is a prime number.

377 isn’t even, and the sum of its digits don’t add up to a multiple of 3. I’ll give it the quick prime number test: 2^377 (mod 377) = 188, not 2, so 377 is definitely a composite number. Scroll down past the puzzle to see its factoring information.

 

 Print the puzzles or type the factors on this excel file: 10 Factors 2015-02-02

• 377 is a composite number.
• Prime factorization: 377 = 13 x 29
• 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 377 has exactly 4 factors.
• Factors of 377: 1, 13, 29, 377
• Factor pairs: 377 = 1 x 377 or 13 x 29
• 377 has no square factors that allow its square root to be simplified. √377 ≈ 19.416

13 and 29 are both Pythagorean triple hypotenuses so 377 can be written as the sum of two squares two different ways, and it is the hypotenuse of FOUR Pythagorean triples, half of which are primitives:

• 19² + 4² = 377
• 16² + 11² = 377
• 135² + 352² = 377²
• 145² + 348² = 377²
• 152² + 345² = 377²
• 260² + 273² = 377²