The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 x 2 = 4. Therefore 299 has exactly 4 factors.
Factors of 299: 1, 13, 23, 299
Factor pairs: 299 = 1 x 299 or 13 x 23
299 has no square factors that allow its square root to be simplified. √299 ≈ 17.292
Warning: The logic needed to solve this Level 4 puzzle is complicated.
The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 x 2 = 4. Therefore 298 has exactly 4 factors.
Factors of 298: 1, 2, 149, 298
Factor pairs: 298 = 1 x 298 or 2 x 149
298 has no square factors that allow its square root to be simplified. √298 ≈ 17.263
Can you fill out this 1 – 12 multiplication table when you’re only given twelve clues?
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.
Prime factorization: 297 = 3 x 3 x 3 x 11 which can be written 297 = 3³ x 11
The exponents in the prime factorization are 3 and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1) = 4 x 2 = 8. Therefore 297 has exactly 8 factors.
Factors of 297: 1, 3, 9, 11, 27, 33, 99, 297
Factor pairs: 297 = 1 x 297, 3 x 99, 9 x 33, or 11 x 27
Taking the factor pair with the largest square number factor, we get √297 = (√9)(√33) = 3√33 ≈ 17.234
This multiplication table has only 14 clues. Is that enough to complete it?
Prime factorization: 296 = 2 x 2 x 2 x 37 which can be written 296 = 2³ x 37
The exponents in the prime factorization are 3 and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1) = 4 x 2 = 8. Therefore 296 has exactly 8 factors.
Factors of 296: 1, 2, 4, 8, 37, 74, 148, 296
Factor pairs: 296 = 1 x 296, 2 x 148, 4 x 74, or 8 x 37
Taking the factor pair with the largest square number factor, we get √296 = (√4)(√74) = 2√74 ≈ 17.2046505
Although I came home from the hospital on Sunday, I haven’t felt like spending much time on the computer reading or writing anything. Full recovery will be another five weeks, but hopefully I’ll start feeling better in less time. I do feel better today than I did on Sunday.
The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 x 2 = 4. Therefore 295 has exactly 4 factors.
Factors of 295: 1, 5, 59, 295
Factor pairs: 295 = 1 x 295 or 5 x 59
295 has no square factors that allow its square root to be simplified. √295 ≈ 17.176
Prime factorization: 294 = 2 x 3 x 7 x 7, which can be written 294 = 2 x 3 x (7^2)
The exponents in the prime factorization are 1, 1 and 2. Adding one to each and multiplying we get (1 + 1)(1 + 1)(2 + 1) = 2 x 2 x 3 = 12. Therefore 294 has exactly 12 factors.
Factor pairs: 294 = 1 x 294, 2 x 147, 3 x 98, 6 x 49, 7 x 42, or 14 x 21
Taking the factor pair with the largest square number factor, we get √294 = (√6)(√49) = 7√6 ≈ 17.146
Sum-Difference Puzzles:
6 has two factor pairs. One of them adds up to 5, and the other one subtracts to 5. Put those factor pairs in the correct boxes to solve the first puzzle.
294 has six factor pairs. Of of those factor pairs adds up to 35, and a different one subtracts to 35. Identify those factor pairs, and you will be able to solve the second puzzle.
The second puzzle is really just the first puzzle in disguise.
Prime factorization: 292 = 2 x 2 x 73, which can be written 292 = 2² x 73
The exponents in the prime factorization are 2 and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1) = 3 x 2 = 6. Therefore 292 has exactly 6 factors.
Factors of 292: 1, 2, 4, 73, 146, 292
Factor pairs: 292 = 1 x 292, 2 x 146, or 4 x 73
Taking the factor pair with the largest square number factor, we get √292 = (√4)(√73) = 2√73 ≈ 17.088.
Can you fill out this multiplication table that has only ten clues?
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.
The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 x 2 = 4. Therefore 291 has exactly 4 factors.
Factors of 291: 1, 3, 97, 291
Factor pairs: 291 = 1 x 291 or 3 x 97
291 has no square factors that allow its square root to be simplified. √291 ≈ 17.059
Can you complete this multiplication table if you’re given just twelve clues?
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 x 2 x 2 = 8. Therefore 290 has 8 factors.
Factors of 290: 1, 2, 5, 10, 29, 58, 145, 290
Factor pairs: 1 x 290, 2 x 145, 5 x 58, or 10 x 29
290 has no square factors that allow its square root to be simplified. √290 ≈ 17.029
