1259 Graveyard Marker

This is the first of a week’s worth of Halloween Find the Factors puzzles. Graveyards are often associated with the holiday. Many graveyards have crosses marking the place where some dearly loved person was laid to rest. This puzzle isn’t very scary. Have fun solving it!

Print the puzzles or type the solution in this excel file: 10-factors-1259-1270

Now I’ll share some facts about the number 1259:

  • 1259 is a prime number.
  • Prime factorization: 1259 is prime.
  • The exponent of prime number 1259 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1259 has exactly 2 factors.
  • Factors of 1259: 1, 1259
  • Factor pairs: 1259 = 1 × 1259
  • 1259 has no square factors that allow its square root to be simplified. √1259 ≈ 35.48239

How do we know that 1259 is a prime number? If 1259 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1259 ≈ 35.5. Since 1259 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1259 is a prime number.

1259 is the sum of the twenty-five prime numbers from 5 to 103.

The number after 1259 has thirty-six factors. No wonder 1259 had to settle for 1 and itself being its only factors.

Between prime numbers 1237 and 1277, there are 39 numbers but only two of them are prime numbers. 1259 is one of them. Up to 1277 on the number line, no other segment of the same length has a lower incidence of prime numbers than that!

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1221 and Level 1

This puzzle is like a multiplication table with its factors in a different order. Can you figure out where the factors from 1 to 10 go in both the first column and the top row? Afterward, can you correctly fill in every cell of this mixed-up multiplication table?

Print the puzzles or type the solution in this excel file: 10-factors-1221-1231

Let me share some facts about the number 1221:

  • 1221 is a composite number.
  • Prime factorization: 1221 = 3 × 11 × 37
  • The exponents in the prime factorization are 1, 1, and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 = 8. Therefore 1221 has exactly 8 factors.
  • Factors of 1221: 1, 3, 11, 33, 37, 111, 407, 1221
  • Factor pairs: 1221 = 1 × 1221, 3 × 407, 11 × 111, or 33 × 37
  • 1221 has no square factors that allow its square root to be simplified. √1221 ≈ 34.94281

1 × 11 × 111 = 1221

1221 is the sum of five consecutive prime numbers:
233 + 239 + 241 + 251 + 257 = 1221

1221 is the hypotenuse of a Pythagorean triple:
396-1155-1221 which is 33 times (12-35-37)

Not only is 1221 a palindrome in base 10 but look at it in these other bases:
It’s 14341 in BASE 5,
5353 in BASE 6,
272 in BASE 23, and
it’s XX in BASE 36 because 33(36) + 33(1) = 33(37) = 1221

1211 and Level 1

Today we are reminded that the world can be a very complicated place. Today’s puzzle isn’t the least bit complicated. Just write the numbers from 1 to 12 in the first column and the top row so that the puzzle looks like a multiplication table (but with the factors not in their usual places.) Afterward, you can fill in the rest of the table. You can do this!

Print the puzzles or type the solution in this excel file: 12 factors 1211-1220

Now I’d like to share some information about the number 1211:

  • 1211 is a composite number.
  • Prime factorization: 1211 = 7 × 173
  • The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 1211 has exactly 4 factors.
  • Factors of 1211: 1, 7, 173, 1211
  • Factor pairs: 1211 = 1 × 1211 or 7 × 173
  • 1211 has no square factors that allow its square root to be simplified. √1211 ≈ 34.79943

1211 is the sum of the seventeen prime numbers from 37 to 107.
It is also the sum of seven consecutive primes:
157 + 163 + 167 + 173 + 179 + 181 + 191 = 1211

1211 is the hypotenuse of a Pythagorean triple:
364-1155-1211 which is 7 times (52-165-173)

1187 and Level 1

What is the biggest number that can divide all the clues in today’s puzzle without leaving a remainder? If you can answer that question, then you also know the greatest common factor of all those clues. It really is that simple. You can solve this puzzle!

Print the puzzles or type the solution in this excel file: 12 factors 1187-1198

Now I’ll share some information about the number 1187:

  • 1187 is a prime number.
  • Prime factorization: 1187 is prime.
  • The exponent of prime number 1187 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1187 has exactly 2 factors.
  • Factors of 1187: 1, 1187
  • Factor pairs: 1187 = 1 × 1187
  • 1187 has no square factors that allow its square root to be simplified. √1187 ≈ 34.45287

How do we know that 1187 is a prime number? If 1187 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1187 ≈ 34.5. Since 1187 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1187 is a prime number.

1187 is the sum of the nineteen prime numbers from 23 to 103.
It is also the sum of three consecutive primes:
389 + 397 + 401 = 1187

1174 and Level 1

I’ve given you just nine clues in this puzzle, but that’s enough to find all the factors AND complete the entire table. I’m serious. I really have given you sufficient information to find the one and only solution to this puzzle!

Print the puzzles or type the solution in this excel file: 10-factors-1174-1186

Now I’ll share some information about the number 1174:

  • 1174 is a composite number.
  • Prime factorization: 1174 = 2 × 587
  • The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 1174 has exactly 4 factors.
  • Factors of 1174: 1, 2, 587, 1174
  • Factor pairs: 1174 = 1 × 1174 or 2 × 587
  • 1174 has no square factors that allow its square root to be simplified. √1174 ≈ 34.26368

1174 is also the sum of the sixteen prime numbers from 41 to 107.

 

1161 and Level 1

Solving this puzzle will help you review the multiplication table. Knowing the multiplication table inside and out will be a big PLUS in your life. It will save you so much time in all your mathematics classes!

Print the puzzles or type the solution in this excel file: 12 factors 1161-1173

Here is some information about the number 1161:

  • 1161 is a composite number.
  • Prime factorization: 1161 = 3 × 3 × 3 × 43, which can be written 1161 = 3³ × 43
  • The exponents in the prime factorization are 3 and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1) = 4 × 2 = 8. Therefore 1161 has exactly 8 factors.
  • Factors of 1161: 1, 3, 9, 27, 43, 129, 387, 1161
  • Factor pairs: 1161 = 1 × 1161, 3 × 387, 9 × 129, or 27 × 43
  • Taking the factor pair with the largest square number factor, we get √1161 = (√9)(√129) = 3√129 ≈ 34.07345

1161 is the sum of the first twenty-six prime numbers. That’s all the primes from 2 to 101.

1161 is a palindrome in a couple of bases:
It’s 10010001001 in BASE 2 because 2¹⁰ + 2⁷ + 2³ + 2⁰ = 1161 and
1B1 in BASE 29 (B is 11 base 10) because 29² + 11(29) + 1 = 1161

1148 and Level 1

This level 1 puzzle has only one solution, and I’m sure you can find it. Just write the numbers from 1 to 10 in both the first column and the top row so that the clues and those numbers make a valid multiplication table.

Print the puzzles or type the solution in this excel file: 10-factors-1148-1160

Now here’s some information about the number 1148:

  • 1148 is a composite number.
  • Prime factorization: 1148 = 2 × 2 × 7 × 41, which can be written 1148 = 2² × 7 × 41
  • 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 × 2 × 2 = 12. Therefore 1148 has exactly 12 factors.
  • Factors of 1148: 1, 2, 4, 7, 14, 28, 41, 82, 164, 287, 574, 1148
  • Factor pairs: 1148 = 1 × 1148, 2 × 574, 4 × 287, 7 × 164, 14 × 82, or 28 × 41
  • Taking the factor pair with the largest square number factor, we get √1148 = (√4)(√287) = 2√287 ≈ 33.88215

1148 is the sum of two consecutive primes:
571 + 577 = 1148

1148 is the hypotenuse of a Pythagorean triple:
252-1120-1148 which is 28 times (9-40-41)

1148 looks interesting to be in these other bases:
It’s 1120112 in BASE 3 because 3⁶ + 3⁵ + 2(3⁴) + 3² + 3¹ + 2(3⁰) = 1148,
1515 in BASE 9 because 9³ + 5(9²) + 9 + 5(1) = 1148,
and 161 in BASE 31 because 31² + 6(31) + 1 = 1148

1121 and Level 1

If you’ve learned how to multiply and divide, then you can solve this puzzle. Just write the numbers from 1 to 10 in both the first column and the top row so that the clues and those factors make a multiplication table. You can definitely do this one!

Print the puzzles or type the solution in this excel file: 10-factors-1121-1133

Here are a few facts about the number 1121:

  • 1121 is a composite number.
  • Prime factorization: 1121 = 19 × 59
  • The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 1121 has exactly 4 factors.
  • Factors of 1121: 1, 19, 59, 1121
  • Factor pairs: 1121 = 1 × 1121 or 19 × 59
  • 1121 has no square factors that allow its square root to be simplified. √1121 ≈ 33.48134

1121 is a palindrome in two other bases:
It’s 1C1 in BASE 28 (C is 12 base 10) because 28² + 12(28) + 1 = 1121,
and it’s 131 in BASE 32 because 32² + 3(32) + 1 = 1121

1111 and Level 1

This is puzzle number 1111, a number made with four 1’s. The puzzle number doesn’t usually have anything to do with the puzzle, but I made an exception this time:  One of the factors of 1111 is important in solving this particular level 1 puzzle. Have fun solving it!

Print the puzzles or type the solution in this excel file: 12 factors 1111-1119

Here are a few things I’ve learned about the number 1111:

  • 1111 is a composite number.
  • Prime factorization: 1111 = 11 × 101
  • The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 1111 has exactly 4 factors.
  • Factors of 1111: 1, 11, 101, 1111
  • Factor pairs: 1111 = 1 × 1111 or 11 × 101
  • 1111 has no square factors that allow its square root to be simplified. √111133.3316666

1111 is the hypotenuse of exactly one Pythagorean triple:
220-1089-1111 which is 11 times (20-99-101)

1111² = 1234321, a very special palindrome!

1111 is a repdigit in base 10, and it is a palindrome in three consecutive bases plus one more:
It’s 787 in BASE 12 because 7(144) + 8(12) + 7(1) = 1111,
676 in BASE 13 because 6(169) + 7(13) + 6(1) = 1111,
595 in BASE 14 because 5(196) + 9(14) + 5(1) = 1111, and it’s
171 in BASE 30 because 1(900) + 7(30) + 1(1) = 1111

 

1102 and Level 1

Write each number from 1 to 10 in both the first column and the top row so that those numbers are the factors of the given clues. This one is not too difficult, so if you haven’t solved one of these puzzles before, give it a try!

Print the puzzles or type the solution in this excel file: 10-factors-1102-1110

Here is some information about the number 1102:

  • 1102 is a composite number.
  • Prime factorization: 1102 = 2 × 19 × 29
  • The exponents in the prime factorization are 1, 1, and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 = 8. Therefore 1102 has exactly 8 factors.
  • Factors of 1102: 1, 2, 19, 29, 38, 58, 551, 1102
  • Factor pairs: 1102 = 1 × 1102, 2 × 551, 19 × 58, or 29 × 38
  • 1102 has no square factors that allow its square root to be simplified. √1102 ≈ 33.19639

1102 = 2(29)(19) so we know that 480-1102-1202 is a Pythagorean triple
calculated from 29²-19², 2(29)(19), 29²+19²

1102 is also the hypotenuse of a Pythagorean triple:
760-798-1102 which is (20-21-29) times 38.

1102 is a palindrome when it is written in a couple of other bases:
It’s 2F2 in BASE 20 (F is 15 base 10) because 2(20²) + 15(20) + 2(1) = 1102,
and it’s 262 in BASE 22 because 2(22²) + 6(22) + 2(1) = 1102.