# 1535 Your Favorite Candy Bar

### Today’s Puzzle:

If you could have your favorite candy bar in the largest size possible, would it actually turn out to be more than you can chew? Of course not. Even if it took a long time for you to finish that candy bar, it would not be an impossible challenge.

Think of this Challenge puzzle the same way. I think this one uses some delightful logic. Find that logic, take your time, and enjoy the process! Print the puzzles or type the solution in this excel file: 10 Factors 1526-1535

### Factors of 1535:

• 1535 is a composite number.
• Prime factorization: 1535 = 5 × 307.
• 1535 has no exponents greater than 1 in its prime factorization, so √1535 cannot be simplified.
• The exponents in the prime factorization are 1 and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 1535 has exactly 4 factors.
• The factors of 1535 are outlined with their factor pair partners in the graphic below. ### Another Fact about the number 1535:

1535 is the hypotenuse of a Pythagorean triple:
921-1228-1535 which is (3-4-5) times 307.

# 1533 Double Double Toil and Trouble

### Today’s Puzzle:

Solving this cauldron-looking mystery level puzzle might be double-double toil and trouble.  Why? There are four sets of clues that have more than one possible common factor. Nevertheless, the puzzle has only one solution, and you can find it if you let logic and a basic multiplication table be your guide. ### Factors of 1533:

Double Double Toil and Trouble… You can use two divisibility tricks to find the factors of 1533, and the second trick uses doubling.

Is 1533 divisible by 3? The trick for finding numbers divisible by 3 is to add up their digits. If the sum is divisible by 3, so is the original number:
1 + 5 + 3 + 3 = 12, so 1533 is divisible by 3.
It really is! See what I mean: 1533 ÷ 3 = 511.

Is it divisible by 7? Look at the previous quotient, 511. Separate the last digit from the rest and subtract its double from the first part: 51 – 2 = 49, so 511 is divisible by 7, and its multiple, 1533, is also divisible by 7.

(We could have used the same doubling trick TWICE on 1533, but that definitely would have been double-double, toil, and trouble. Just dividing 1533 by 7 to get 219 would be easier!)

• 1533 is a composite number.
• Prime factorization: 1533 = 3 × 7 × 73.
• 1533 has no exponents greater than 1 in its prime factorization, so √1533 cannot be simplified.
• The exponents in the prime factorization are 1, 1, and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 = 8. Therefore 1533 has exactly 8 factors.
• The factors of 1533 are outlined with their factor pair partners in the graphic below. ### More Facts About the Number 1533:

1533 is the hypotenuse of a Pythagorean triple, and you can apply those same two divisibility tricks to all three numbers in that triple. (Note: 3 × 7 = 21)
1008-1155-1533 which is 21 times (48-55-73)

1533 is the difference of two squares in FOUR different ways:
767² – 766² = 1533,
257² – 254² = 1533,
113² – 106² = 1533, and
47² – 26² = 1533.

# 1532 Don’t Let This Puzzle Spook You!

### Today’s Puzzle:

If this little ghost prowls your neighborhood this Halloween, don’t let it spook you. Sure, it is a level 6 puzzle, but if you stick to using logic from start to finish, you’ll know the most about this ghost! Here’s the same puzzle minus the embellishments: ### Factors of 1532:

• 1532 is a composite number.
• Prime factorization: 1532 = 2 × 2 × 383, which can be written 1532 = 2² × 383
• 1532 has at least one exponent greater than 1 in its prime factorization so √1532 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1532 = (√4)(√383) = 2√383
• The exponents in the prime factorization are 2 and 1. Adding one to each exponent and multiplying we get (2 + 1)(1 + 1) = 3 × 2 = 6. Therefore 1532 has exactly 6 factors.
• The factors of 1532 are outlined with their factor pair partners in the graphic below. ### More about the Number 1532:

1532 = 2 × 383 × 2. That factorization looks the same frontwards or backward.

1532 can be written as the difference of two squares:
384² – 382² = 1532.

# 1530 Jack-o’-lantern

### Today’s Puzzle:

Here’s a Jack-O’-Lantern Puzzle for you to enjoy. It’s a Level 5 puzzle so it might be more of a trick than a treat. Remember to use logic every step of the way instead of guessing and checking. Here’s the same puzzle without any added color: ### Factors of 1530:

15 is half of 30, so 1530 is divisible by 6 just like all these numbers are divisible by 6: 12, 24, 36, 48, 510, 612, 714, 816, 918, 1020, 1122, 1224, 1326, 1428, and so forth.

• 1530 is a composite number.
• Prime factorization: 1530 = 2 × 3 × 3 × 5 × 17, which can be written 1530 = 2 × 3² × 5 × 17
• 1530 has at least one exponent greater than 1 in its prime factorization so √1530 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1530 = (√9)(√170) = 3√170
• The exponents in the prime factorization are 1, 2, 1, and 1. Adding one to each exponent and multiplying we get (1 + 1)(2 + 1)(1 + 1)(1 + 1) = 2 × 3 × 2 × 2 = 24. Therefore 1530 has exactly 24 factors.
• The factors of 1530 are outlined with their factor pair partners in the graphic below. ### More about the Number 1530:

51 × 30 = 1530. Did you notice that the same digits appear on both sides of the equal sign and only +, -, ×, ÷, (), or exponents were used to make a true statement? 1530 is only the 25th number that can make that claim, so we call it the 25th Friedman number.

There are MANY possible factor trees for 1530, but let’s celebrate that it is also a Friedman number with this one: 1530 is the hypotenuse of FOUR Pythagorean triples:
234-1512-1530, which is 18 times (13-84-85),
648-1386-1530, which is 18 times (36-77-85),
720-1350-1530, which is (8-15-17) times 90, and
918-1224-1530, which is (3-4-5) times 306.

# 1529 Pointy Hat

#### Today’s Puzzle:

A pointy hat is part of many different Halloween costumes: Wizards, Witches, Medieval Princesses, and Clowns come to my mind. Today’s puzzle looks like a pointy hat. Use logic to work your magic in solving it! As always, there is only one solution. ### Factors of 1529:

Look at this math fact using the digits of 1529:
1 – 5 + 2 – 9 = -11, a number divisible by 11, so 1529 is divisible by 11.

• 1529 is a composite number.
• Prime factorization: 1529 = 11 × 139.
• 1529 has no exponents greater than 1 in its prime factorization, so √1529 cannot be simplified.
• The exponents in the prime factorization are 1 and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 1529 has exactly 4 factors.
• The factors of 1529 are outlined with their factor pair partners in the graphic below. ### More about the Number 1529:

1529 is the difference of two squares in two different ways:
765² – 764²  = 1529, and
75² – 64²  = 1529.

# 1528 Candy Corn

### Today’s Puzzle:

To solve this Level 3 Candy Corn Halloween puzzle, first, find the factors that will work with the clues in the top and bottom rows. Then work you way down row by row filling in factors as you go.

When you get to the 8 in this puzzle, will the factors be 8 × 1 or 4 × 2? Two of those factors will be eliminated because they already appear in the first column. The other two remain possibilities, but one of those factors cannot appear in any other place in that first column, so that is the one you will want to choose. Have a sweet time solving this puzzle! Here is a plain version of the same puzzle: ### Factors of 1528:

1528 is divisible by two because it is even.

1528 is divisible by four because its last two digits (in the same order) make a number, 28, which is divisible by 4.

Can 1528 be evenly divided by 8? Yes. Here’s a quick way to know: 28 is divisible by 4, but not by 8, AND 5 is odd, so 1528 is divisible by 8, as is every other number ending in 528.

• 1528 is a composite number.
• Prime factorization: 1528 = 2 × 2 × 2 × 191, which can be written 1528 = 2³ × 191
• 1528 has at least one exponent greater than 1 in its prime factorization so √1528 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1528 = (√4)(√382) = 2√382
• The exponents in the prime factorization are 3 and 1. Adding one to each exponent and multiplying we get (3 + 1)(1 + 1) = 4 × 2 = 8. Therefore 1528 has exactly 8 factors.
• The factors of 1528 are outlined with their factor pair partners in the graphic below. ### Another Fact about the Number 1528:

Since 1528 is divisible by 8 but not by 16, it can be written as the sum of 16 consecutive numbers:
88+89+90+91+92+93+94+95+96+97+98+99+100+101+102+103=1528.

Note that 95 + 96 = 191, and 8 × 191 = 1528.
Likewise, 94 + 97 = 1528,
93 + 98 = 1528, and so forth until we get to…
88 + 103 = 1528.

# 1527 Not a Haunted House

### Today’s Puzzle:

We see plenty of pumpkins on doorsteps this time of year, so I put a few in this puzzle. The puzzle looks a bit like a house, but certainly not a haunted house. Can you write the numbers from 1 to 10 in both the first column and the top row so that the given number clues are the products of those numbers?

Here’s the same puzzle without the added colorful embellishments: ### Factors of 1527:

• 1527 is a composite number.
• Prime factorization: 1527 = 3 × 509
• 1527 has no exponents greater than 1 in its prime factorization, so √1527 cannot be simplified.
• The exponents in the prime factorization are 1, and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 1527 has exactly 4 factors.
• The factors of 1527 are outlined with their factor pair partners in the graphic below. ### Another Fact about the Number 1527:

1527 is the hypotenuse of a Pythagorean triple:
660-1377-1527, which is 3 times (220-459-509).

# 1526 Grave Marker

### Today’s Puzzle:

It’s almost Halloween! This is my favorite kind of grave marker, one that is really just a Find the Factors puzzle in disguise. It’s only a level one, so it isn’t very tricky. I hope you find it a real treat! Here’s the same puzzle but requiring less ink to print: Neighbors have decorated part of their yard to look like a mini graveyard for Halloween. I think my grave marker would fit right in! ### Factors of 1526:

• 1526 is a composite number.
• Prime factorization: 1526 = 2 × 7 × 109.
• 1526 has no exponents greater than 1 in its prime factorization, so √1526 cannot be simplified.
• The exponents in the prime factorization are 1, 1, and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 = 8. Therefore 1526 has exactly 8 factors.
• The factors of 1526 are outlined with their factor pair partners in the graphic below. ### Another Fact about the Number 1526:

1526 is the hypotenuse of a Pythagorean triple:
840-1274-1526 which is 14 times (60-91-109)

# 1524 Mystery

### Today’s Puzzle:

Is this puzzle easy or difficult? That is the mystery. Will any of the clues trick you into writing factors that won’t work with the rest of the puzzle? If you consistently use logic and not just multiplication and division facts, you’ll solve this puzzle. ### Factors of 1524:

• 1524 is a composite number.
• Prime factorization: 1524 = 2 × 2 × 3 × 127, which can be written 1524 = 2² × 3 ×
• 1524 has at least one exponent greater than 1 in its prime factorization so √1524 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1524 = (√4)(√381) = 2√
• The exponents in the prime factorization are 2, 1, and 1. Adding one to each exponent and multiplying we get (2 + 1)(1 + 1)(1 + 1) = 3 × 2 × 2 = 12. Therefore 1524 has exactly 12 factors.
• The factors of 1524 are outlined with their factor pair partners in the graphic below. ### More about the Number 1524:

1524 is the difference of two squares in two different ways:
382² – 380² = 1524,
130² – 124² = 1524.

# 1523 Mystery Puzzle

### Today’s Puzzle:

My newly turned 10-year-old granddaughter likes making Find the Factors 1 -12 Puzzles with me. Unfortunately, at the beginning of the month, she was in an accident. Her 12-year-old cousin hadn’t ever made a puzzle before but helped me make this one to wish her a speedy recovery. (Thankfully, she is almost fully recovered now.) ### Factors of 1523:

• 1523 is a prime number.
• Prime factorization: 1523 is prime.
• 1523 has no exponents greater than 1 in its prime factorization, so √1523 cannot be simplified.
• The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1523 has exactly 2 factors.
• The factors of 1523 are outlined with their factor pair partners in the graphic below. How do we know that 1523 is a prime number? If 1523 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1523. Since 1523 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 31, or 37, we know that 1523 is a prime number.

### Another Fact about the Number 1523:

1523 is the difference of two consecutive squares:
762² – 761² = 1523.