Problems Related to the Properties of Numbers

 

When Macdonald deals with what he refers to as numbers in this section of the notebook, he is working with the positive integers 1, 2, 3, 4, etc.

 

He writes out four problems in this section.

 

Problem 1: To find the divisor a number.

Solution: Proceed by trial and error.

 

Problem 2: To find the greatest common divisor of a set of numbers. This is the largest number that divides each of the numbers in the set.

Solution: The method described is known as Euclid’s algorithm since it was given in Book 7, Propositions 1 and 2 of Euclid’s Elements. Macdonald provides only the algorithm and makes no reference to Euclid.

 

For two numbers, divide the smaller into the greater. This yields a divisor times a number plus a remainder. Divide this divisor by the remainder. Repeat the procedure until the numbers divide evenly. The last divisor is the greatest common divisor.

 

Example: Macdonald finds the greatest common divisor of 475 and 589.

589 = 1 × 475 + 114

475 = 4 × 114 + 19

114 = 6 × 19

Therefore 19 is the greatest common divisor (475/19 = 26 and 589/19 = 31).

 

For three numbers, find the greatest common divisor of two of them; call it . Then find the greatest common divisor of  and the third number. The result is the greatest common divisor of all three numbers.

 

Example: Macdonald finds the greatest common divisor of 1080, 945, and 747.

1080 = 1 × 945 + 135

945 = 7 × 135

Therefore 135 is the greatest common divisor of 1080 and 945.

747 = 5 × 135 + 72

135 = 1 × 72 + 63

72 = 1 × 63 + 9

63 = 7 × 9

Therefore 9 is the greatest common divisor of 1080, 945, and 747 (1080/9 = 120, 945/9 = 105, and 747/9 = 83).

 

Problem 3: To find a common multiple of any set of numbers.

Solution: Multiply the numbers together.

 

Problem 4: To find the least common multiple of any set of numbers. This is the smallest number which is divisible by each of the numbers in the set.

Solution: If the numbers are all prime numbers, then multiply them all together. If some are not prime, express them as products of prime numbers. The least common multiple is the product of all the unique prime numbers.

Example: Macdonald finds the least common multiple of 3, 7, 10, 21, and 30.  

3 and 7 are primes.

The remaining numbers can be factored as: 10 = 2 × 5, 21 = 3 × 7 and 30 = 2 × 3 × 5.

Therefore 2 × 3 × 5 × 7 = 210 is the least common multiple.

 

At this point Macdonald errs and calls the least common multiple the least common divisor. By not being careful about what he was copying, this is the first occasion which hints that Macdonald’s interest in mathematics may have been limited.

 

One of the leaves of the notebook is missing at this point and this section abruptly ends with the one example.