This level 1 puzzle will help you focus on one set of division facts. You can find all the factors that belong in the first column and the top row if you know those division facts. After you find all the factors from 1 to 10, you can fill in the entire multiplication table.

Print the puzzles or type the solution in this excel file: 10-factors-968-977

970 is the sum of two squares two different ways:
23² + 21² = 970
31² + 3²= 970

That means 970 is the hypotenuse of more than one Pythagorean triple:
88-966-970 calculated from 23² – 21², 2(23)(21), 23² + 21²
186-952-970 calculated from 2(31)(3), 31² – 3², 31² + 3²
582-776-970 which is (3-4-5) times 194
650-720-970 which is 10 times (65-72-97)

Here’s a fun fact: 970 is 202 in BASE 22 because 2(22²) + 2(1) = 2(484 + 1) = 2(485) = 970

• 970 is a composite number.
• Prime factorization: 970 = 2 × 5 × 97
• 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 970 has exactly 8 factors.
• Factors of 970: 1, 2, 5, 10, 97, 194, 485, 970
• Factor pairs: 970 = 1 × 970, 2 × 485, 5 × 194, or 10 × 97
• 970 has no square factors that allow its square root to be simplified. √970 ≈ 31.14482

Whether you are young, old, or in between, if you can do some simple division, then you can solve this level 1 puzzle. There is a column of clues and a row of clues. Both of them have the same common factor. Write that common factor in the first column to the left of the row of clues and again in the top row above the column of clues. Then simply divide. You will be done in no time at all.

Print the puzzles or type the solution in this excel file: 12 factors 959-967

Here are some facts about the number 957:

959 is the hypotenuse of a Pythagorean triple:
616-735-959 which is 7 times (88-105-137)

959 is a palindrome in base 10.

And it is a cool-looking 1110111111 in BASE 2
because (2¹⁰ – 1) – 2⁶ = 959.
In base 2 we would write (if we use commas)
1,111,111,111 – 1,000,000 = 1,110,111,111

• 959 is a composite number.
• Prime factorization: 959 = 7 × 137
• 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 959 has exactly 4 factors.
• Factors of 959: 1, 7, 137, 959
• Factor pairs: 959 = 1 × 959 or 7 × 137
• 959 has no square factors that allow its square root to be simplified. √959 ≈ 30.967725

If you’ve never solved a Find the Factors puzzle before, this one will be perfect for you to do. It only has nine clues, but that is enough to find all the factors and fill in the entire multiplication table. You will feel quite smart when you’re done, too.

Print the puzzles or type the solution in this excel file: 10-factors-951-958

This is my 952nd post so I will mention a few facts about that number.

OEIS.org informs us that 9³ + 5³ + 2³ + 9 × 5 × 2 = 952.

952 is the hypotenuse of a Pythagorean triple:
448-840-952 which is (8-15-17) times 56

952 looks interesting in some other bases:
4224 in BASE 6 because 4(6³) + 2(6²) + 2(6¹) + 4(6⁰) = 952
2C2 in BASE 19 (C is 12 Base 10) because 2(19²) + 12(19¹) +2(19⁰) = 952
SS BASE 33 (S is 28) because 28(33¹) + 28(33⁰) = 28(33 + 1) = 28(34) = 952
S0 BASE 34 because 28(34¹) + 0(34⁰) = 28(34) = 952

• 952 is a composite number.
• Prime factorization: 952 = 2 × 2 × 2 × 7 × 17, which can be written 952 = 2³ × 7 × 17
• The exponents in the prime factorization are 3, 1, and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1)(1 + 1) = 4 × 2 × 2 = 16. Therefore 952 has exactly 16 factors.
• Factors of 952: 1, 2, 4, 7, 8, 14, 17, 28, 34, 56, 68, 119, 136, 238, 476, 952
• Factor pairs: 952 = 1 × 952, 2 × 476, 4 × 238, 7 × 136, 8 × 119, 14 × 68, 17 × 56, or 28 × 34
• Taking the factor pair with the largest square number factor, we get √952 = (√4)(√238) = 2√238 ≈ 29.854497

This puzzle is probably as tough as a level 1 puzzle can get, but don’t let that prevent you from giving it a try! Can you figure out where the factors from 1 to 12 go in both the first column and the top row?

Print the puzzles or type the solution on this excel file: 12 factors 942-950

Now let me tell you a little about the number 942:

It is the sum of four consecutive prime numbers:
229 + 233 + 239 + 241 = 942

It is the hypotenuse of a Pythagorean triple:
510-792-942 which is 6 times (85-132-157)

942 is a palindrome in two other bases and a repdigit in another:
787 in BASE 11, because 7(11²) + 8(11¹) + 7(11⁰) = 942
272 in BASE 20, because 2(20²) + 7(20¹) + 2(20⁰) = 942
666 in BASE 12, because 6(12²) + 6(12¹) + 6(12⁰) = 6(144+12+1) = 6(157) = 942

942³ is 835,896,888. OEIS.org tells us that 942³ is the smallest perfect cube that contains five 8‘s.

• 942 is a composite number.
• Prime factorization: 942 = 2 × 3 × 157
• 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 942 has exactly 8 factors.
• Factors of 942: 1, 2, 3, 6, 157, 314, 471, 942
• Factor pairs: 942 = 1 × 942, 2 × 471, 3 × 314, or 6 × 157
• 942 has no square factors that allow its square root to be simplified. √942 ≈ 30.6920185

915 is the hypotenuse of four Pythagorean triples:

• 165-900-915 which is 15 times (11-60-61)
• 408-819-915 which is 3 times (136-273-305)
• 621-672-915 which is 3 times (207-224-305)
• 549-732-915 which is (3-4-5) times 183

Print the puzzles or type the solution on this excel file: 10-factors-914-922

Here’s a little more about the number 915:

915 is repdigit 555 in BASE 13 because 5(13²) + 5(13) + 5(1) = 5(183) = 915.

It is palindrome 393 in BASE 16 because 3(16²) + 9(16) +3(1) = 915.

And as 195 in BASE 26, it uses its base 10 digits in a different order. Note that 1(26²) + 9(26) + 5(1) = 915.

• 915 is a composite number.
• Prime factorization: 915 = 3 × 5 × 61
• 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 915 has exactly 8 factors.
• Factors of 915: 1, 3, 5, 15, 61, 183, 305, 915
• Factor pairs: 915 = 1 × 915, 3 × 305, 5 × 183, or 15 × 61
• 915 has no square factors that allow its square root to be simplified. √915 ≈ 30.2489669.

905 is the sum of the seventeen prime numbers from 19 to 89.

905 is also the sum of these seven consecutive prime numbers:

• 109 + 113 + 127 + 131 + 137 + 139 + 149 = 905

Here’s today’s puzzle:

Print the puzzles or type the solution on this excel file: 12 factors 905-913

29² + 8² = 905, and 28² + 11² = 905, making 905 the hypotenuse of four Pythagorean triples:

• 95-900-905, which is 5 times (19-180-181)
• 464-777-905, computed from 2(29)(8), 29² – 8², 29² + 8²
• 543-724-905, which is (3-4-5) times 181
• 616-663-905, computed from 2(28)(11), 28² – 11², 28² + 11²

The numbers in red are factors of 905.

• 905 is a composite number.
• Prime factorization: 905 = 5 × 181
• 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 905 has exactly 4 factors.
• Factors of 905: 1, 5, 181, 905
• Factor pairs: 905 = 1 × 905 or 5 × 181
• 905 has no square factors that allow its square root to be simplified. √905 ≈ 30.0832179

(3 + 0)(3 + 10)(3 + 20) = 897

897 is the hypotenuse of a Pythagorean triple:

• 345-828-897, which is (5-12-13) times 69.

Print the puzzles or type the solution on this excel file: 10-factors-897-904

• 897 is a composite number.
• Prime factorization: 897 = 3 × 13 × 23
• 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 897 has exactly 8 factors.
• Factors of 897: 1, 3, 13, 23, 39, 69, 299, 897
• Factor pairs: 897 = 1 × 897, 3 × 299, 13 × 69, or 23 × 39
• 897 has no square factors that allow its square root to be simplified. √897 ≈ 29.949958.

886 is the sum of the sixteen prime numbers from 23 to 89.

886 is a palindrome in a couple of other bases:

• 12021 in BASE 5, because 1(5⁴) + 2(5³) + 0(5²) + 2(5) + 1(1) = 886
• 474 in BASE 14, because 4(14²) + 7(14) + 4(1) = 886

Print the puzzles or type the solution on this excel file: 12 factors 886-896

• 886 is a composite number.
• Prime factorization: 886 = 2 × 443
• 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 886 has exactly 4 factors.
• Factors of 886: 1, 2, 443, 886
• Factor pairs: 886 = 1 × 886 or 2 × 443
• 886 has no square factors that allow its square root to be simplified. √886 ≈ 29.76575