121 and Level 6

2014-05-172020-04-27 / ivasallay / 1 Comment

121 is not a clue in today’s puzzle. However, when 121 is a clue in a FIND THE FACTORS 1 – 12 puzzle, place 11 in the corresponding cells in both the factor row and the factor column.

This week’s puzzles and last week’s factors: 10 Factors 2014-05-12

Here’s more about the number 121:

121 is a composite number.
Prime factorization: 121 = 11 × 11 which can be written 121 = 11²
The exponent in the prime factorization is 2. Adding one we get (2 + 1) = 3. Therefore 121 has exactly 3 factors.
Factors of 121: 1, 11, 121
Factor pairs: 121 = 1 × 121 or 11 × 11
121 is a perfect square. √121 = 11

121 = 11². Those digits 1, 1, 2 make 121 the second Friedman number because they can be arranged as a mathematical expression that equals a number that uses those same digits.

121 is not only a perfect square but it is also a star! In fact, games of Chinese checkers will be based on that perfect star.

Since 121 is the square of 11, the fifth prime number, it is only the fifth number to have exactly 3 factors.

121 is the sum of the three prime numbers from 37 to 43.

121 looks interesting when it is written in some other bases:
It’s repdigit 11111 in BASE 3 because 3⁴ + 3³ + 3² + 3¹ + 3⁰ = 121
and it’s 321 in BASE 6.

Not only is it a palindrome in base 10, but
it’s 232 in BASE 7, and
it’s 171 in BASE 8

It can also look pretty square depending on the base used:
It’s 441 in BASE 5,
144 in BASE 9,
121 in BASE 10 (of course),
100 in BASE 11,
81 in BASE 15, and
49 in BASE 28

121 is a pretty remarkable number!

If you got stuck solving the puzzle above, here’s one logical way to find the solution:

120 and Level 5

2014-05-162021-01-29 / ivasallay / 6 Comments

Today’s Puzzle:

Write the numbers from 1 to 10 in both the top row and the first column so that this puzzle functions like a multiplication table.

Excel file of this week’s puzzles and last week’s factors: 10 Factors 2014-05-12

Thinking process using divisibility rules to find the factor pairs of 120:

√120 is irrational and approximately equal to 10.95. Every factor pair of 120 will have one factor less than 10.95 and one factor greater than 10.95, and we will find both factors in each pair at the same time. The following numbers are less than 10.95. Are they factors of 120?

Yes, all whole numbers are divisible by 1, so 1 x 120 = 120.
Yes, 120 is an even number. 120 ÷ 2 = 60, so 2 x 60 = 120. (Since 60 is even, 4 will also be a factor of 120.)
Yes, 1 + 2 + 0 = 3 which is divisible by 3 (but not by 9), so 120 is divisible by 3. 120 ÷ 3 = 40, so 3 x 40 = 120. Note 120 will not be divisible by 9.
Yes, the number formed from its last two digits, 20, is divisible by 4, so 120 is divisible by 4, and 4 x 30 = 120. (Note since 30 is even, 8 will also be a factor of 120.)
Yes, the last digit is 0 or 5, so 120 is divisible by 5, and 5 x 24 = 120.
Yes, 120 is divisible by both 2 and 3, so it is divisible by 6, and 6 x 20 = 120.
No. The divisibility trick for 7 requires us to split 120 into 12 and 0. We double 0 and subtract the double from 12. 12 – (2 x 0) = 12 – 0 = 12. Since 12 is not divisible by 7, 120 also is not divisible by 7.
Yes, see 4 above. 120 = 8 x 15. (This will mean that ANY number whose last 3 digits are 120 will also be divisible by 8.)
No, see 3 above. 120 is not divisible by 9.
Yes, 120 ends with a zero, so 10 is a factor of 120, and 10 x 12 = 120.

From this thinking process we conclude that the factor pairs of 120 are 1 x 120, 2 x 60, 3 x 40, 4 x 30, 5 x 24, 6 x 20, 8 x 15, and 10 x 12.

Factors of 120:

120 is a composite number. 120 = 1 x 120, 2 x 60, 3 x 40, 4 x 30, 5 x 24, 6 x 20, 8 x 15, or 10 x 12. Factors of 120: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120. Prime factorization: 120 = 2 x 2 x 2 x 3 x 5, which can also be written 120 = 2³ x 3 x 5.

When 120 is a clue in the FIND THE FACTORS 1 – 12 puzzles, use 10 and 12 as the factors.

Sum-Difference Puzzle:

30 has four factor pairs. One of those factor pairs adds up to 13, and another one subtracts to 13. Can you write those factors in their proper places in the first puzzle below?

120 has eight factor pairs. One of those factor pairs adds up to 26, and another one subtracts to 26. 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 120:

120 = 5! because 1·2·3·4·5 = 120

120 is also the smallest positive multiple of 6 that is neither preceded nor followed by a prime number! (119 = 7 ×1 7, and 121 = 11 × 11, so neither one is prime.)

What kind of shape is 120 in?

120 is the 15th triangular number because 15(16)/2 = 120,
it’s the 8th tetrahedral number because (8)(9)(10)/6 = 120 (That means
120 is the sum of the first eight triangular numbers), and
it is the 8th hexagonal number because (8)(2·8-1) = 120.

120 is the hypotenuse of a Pythagorean triple:
72-96-120, which is 24 times (3-4-5).

A Logical Way to Find the Solution to Today’s Puzzle:

119 and Level 4

2014-05-152020-04-27 / ivasallay / Leave a comment

119 is a composite number. 119 = 1 x 119 or 7 x 17. Factors of 119: 1, 7, 17, 119. Prime factorization: 119 = 7 x 17.

119 is never a clue in the FIND THE FACTORS puzzles.

119 is the sum of consecutive prime numbers two ways. Each way begins with one of its prime factors!
7 + 11 + 13 + 17 + 19 + 23 + 29 = 119 (That’s 7 consecutive primes!)
17 + 19 + 23 + 29 + 31 = 119

Excel file of this week’s puzzles and last week’s factors: 10 Factors 2014-05-12

118 and Level 3

2014-05-142020-04-27 / ivasallay / Leave a comment

118 is a composite number. 118 = 1 x 118 or 2 x 59. Factors of 118: 1, 2, 59, 118. Prime factorization: 118 = 2 x 59.

118 is never a clue in the FIND THE FACTORS puzzles.

Excel file of puzzles and the previous week’s factors: 10 Factors 2014-05-12

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:

117 and Level 2

2014-05-132020-04-27 / ivasallay / Leave a comment

117 is a composite number. 117 = 1 x 117, 3 x 39, or 9 x 13. Factors of 117: 1, 3, 9, 13, 39, 117. Prime factorization: 117 = 3 x 3 x 13, which can also be written 3² x 13.

117 is never a clue in the FIND THE FACTORS puzzles.

Excel file of this week’s puzzles and last week’s factors: 10 Factors 2014-05-12

116 and Level 1

2014-05-122020-04-27 / ivasallay / Leave a comment

116 is a composite number. 116 = 1 x 116, 2 x 58, or 4 x 29. Factors of 116: 1, 2, 4, 29, 58, 116. Prime factorization: 116 = 2 x 2 x 29, which can also be written 2² x 29.

116 is never a clue in the FIND THE FACTORS puzzles.

Excel file of this week’s puzzles and last week’s factors: 10 Factors 2014-05-12

115 and Level 6

2014-05-102020-04-27 / ivasallay / Leave a comment

115 is a composite number. 115 = 1 x 115 or 5 x 23. Factors of 115: 1, 5, 23, 115. Prime factorization: 115 = 5 x 23.

115 is never a clue in the FIND THE FACTORS puzzles.

Excel file of puzzles and the previous week’s solutions: 12 Factors 2014-05-05

114 My daughter – a new wife with a new life

2014-05-092020-04-27 / ivasallay / 5 Comments

114 is a composite number. 114 = 1 x 114, 2 x 57, 3 x 38, or 6 x 19. Factors of 114: 1, 2, 3, 6, 19, 38, 57, 114. Prime factorization: 114 = 2 x 3 x 19.

114 is never a clue in the FIND THE FACTORS puzzles.

My recently married daughter and her husband will be moving to London in June where he will be employed by Medopad. In the meantime he has an assignment to get Likes on Medopad’s facebook page, so he is sponsoring a contest until May 22, 2014 for people living in the United States, Great Britian, or Germany for a $100 Visa gift card. To enter all you have to do is click the link below and follow the instructions. If you also invite your Facebook friends to like Medopad and they fill out the application, not only does it enter them in the contest, but it also increases YOUR chances of winning as well. If you live outside those 3 countries, you can still like Medopad’s facebook page and Medopad will contribute $5 for every new like towards a mHealth Impact Foundation that saves lives.

Click https://a.pgtb.me/Qx0cNZ to like. Visit www.medopad.com to learn more about this revolutionary company.

Update: The contest was won by Kathleen Bundy. My son-in-law was able to get plenty of Likes on the Medopad’s facebook page to complete his assignment.