| 172 is a composite number. |
| Factor pairs: 172 = 1 x 172, 2 x 86, or 4 x 43 |
| Factors of 172: 1, 2, 4, 43, 86, 172 |
| Prime factorization: 172 = 2 x 2 x 43, which can be written (2^2) x 43 |
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Excel file of puzzles and previous week’s factor solutions: 10 Factors 2014-07-07
| 171 is a composite number. |
| Factor pairs: 171 = 1 x 171, 3 x 57, or 9 x 19 |
| Factors of 171: 1, 3, 9, 19, 57, 171 |
| Prime factorization: 170 = 3 x 3 x 19, which can be written 170 = (3^2) x 19 |
171 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 + 17 + 18 and is therefore a triangular number.
The fact that 171 is a triangular number is easy to recognize because 170 = 9 x 19 = (18/2) x 19. Any integer resulting from n(n + 1)/2 is a triangular number.
Excel file of puzzles and previous week’s factor solutions: 10 Factors 2014-07-07
| 170 is a composite number. |
| Factor pairs: 170 = 1 x 170, 2 x 85, 5 x 34, or 10 x 17 |
| Factors of 170: 1, 2, 5, 10, 17, 34, 85, 170 |
| Prime factorization: 170 = 2 x 5 x 17 |
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Excel file of puzzles and previous week’s factor solutions: 10 Factors 2014-07-07
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.
| 169 is a composite number and a perfect square. |
| Factor pairs: 169 = 1 x 169 or 13 x 13 |
| Factors of 169: 1, 13, 169 |
| Prime factorization: 169 = 13 x 13 which can be written 13² |
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Excel file of puzzles and previous week’s factor solutions: 10 Factors 2014-07-07
Nerdinthebrain is a very well-rounded and inspiring blogger. I feel quite honored that she nominated me for the Very Inspiring Blogger Award.
Just two little rules for accepting this award:
1) The nominee shall display the Very Inspiring Blogger Award logo on her/his blog, and link to the blog they got nominated from.
2) The nominee shall nominate fifteen (15) bloggers she/he admires, by linking to their blogs and informing them about it.
Because this award has these requirements, it may seem like a modern version of a chain letter, but it also appears to be a great way to step out of our comfort zones. It helps us read posts and share ideas with people with whom we have a little something in common but just don’t know it yet. Here are my 15 nominees:
Congratulations to each of you. (I’ll do the informing tomorrow because I’ve already spent more time than usual on the internet today.)
√168 ≈ 12.96148. Let’s divide 168 by each number from 1 to 12 to find its factor pairs.
The prime factorization of 168 is 2³ × 3 × 7.
Adding 1 to each of the exponents in the prime factorization and multiplying, we get
(3 + 1)(1 + 1)(1 + 1) = 4 × 2 × 2 = 12. Notice that 168 has exactly 12 factors.
Four of those factor pairs are made up of only even numbers, so 168 is the difference of two squares four different ways:
43² – 41² = 168,
23² – 19² = 168,
17² – 11² = 168,
13² – 1² = 168.
Since 168 is 3 × 56, it is the sum of three consecutive numbers with 56 as the middle number:
55 + 56 + 57 = 168.
Since 168 is 7 × 24, it is the sum of seven consecutive numbers with 24 as the middle number:
21 + 22 + 23 + 24 + 25 + 26 + 27 = 168.
Finally, since 168 is divisible by 8, but not by 16, it is the sum of 16 consecutive numbers:
3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 + 17 + 18 = 168.
As a consequence of that last mathematical fact, here’s another way to make 168:
18² – 17² + 16² – 15² + 14² – 13² + 12² – 11² + 10² – 9² + 8² – 7² + 6² – 5² + 4² – 3² = 168.
I bet you weren’t expecting that!
168 = 6 × 28, so 168 is the product of the first two perfect numbers! Why are those numbers perfect? Each of them is the sum of their divisors:
6 = 1 + 2 + 3, and
28 = 1 + 2 + 4 + 7 + 14.
168 is a repdigit in several other bases:
It’s CC in base 13 because 12(13+1) = 168,
88 in base 20 because 8(20+1) = 168,
77 in base 23 because 7(23+1) = 168,
66 in base 27 because 6(27+1) = 168,
44 in base 41 because 4(41+1) = 168,
33 in base 55 because 3(55+1) = 168,
22 in base 83 because 2(83+1) = 168, and
11 in base 167 because 1(167+1) = 168.
How do we know that 167 is a prime number? If 167 were not a prime number, then it would be divisible by at least one prime number less than or equal to √167 ≈ 12.9. Since 167 cannot be divided evenly by 2, 3, 5, 7, or 11, we know that 167 is a prime number.
Excel file of puzzles and previous week’s factor solutions: 10 Factors 2014-07-07
| 166 is a composite number. |
| Factor pairs: 166 = 1 x 166 or 2 x 83 |
| Factors of 166: 1, 2, 83, 166 |
| Prime factorization: 166 = 2 x 83 |
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Excel file of puzzles and previous week’s factor solutions: 12 Factors 2014-06-30
| 165 is a composite number. |
| Factor pairs: 165 = 1 x 165, 3 x 55, 5 x 33, 11 x 15 |
| Factors of 165: 1, 3, 5, 11, 15, 33, 55, 165 |
| Prime factorization: 165 = 3 x 5 x 11 |
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Excel file of puzzles and previous week’s factor solutions: 12 Factors 2014-06-30
| 164 is a composite number. |
| Factor pairs: 164 = 1 x 164, 2 x 82, or 4 x 41 |
| Factors of 164: 1, 2, 4, 41, 82, 164 |
| Prime factorization: 164 = 2 x 2 x 41, which can be written 164 = (2^2) x 4 |
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Excel file of puzzles and previous week’s factor solutions: 12 Factors 2014-06-30
How do we know that 163 is a prime number? If 163 were not a prime number, then it would be divisible by at least one prime number less than or equal to √163 ≈ 12.8. Since 163 cannot be divided evenly by 2, 3, 5, 7, or 11, we know that 163 is a prime number.
Excel file of puzzles and previous week’s factor solutions: 12 Factors 2014-06-30
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.
