General statements

The idea

A statement describing a pattern or property that holds in all possible cases is a general statement, and the process of arriving at it is generalisation; it can be described using algebra.

Many pairs, one description

The HCF of 6 and 18 is 6. The HCF of 7 and 35 is 7. The HCF of 12 and 60 is 12. All these pairs follow the same pattern.

How could we describe all such pairs in one go?

• With a letter-number, like n and 5n
• By listing every pair
• It cannot be done

Test a pattern on four numbers at once

The first number is a letter-number. Build the other number. The table tries four values of the letter at once. Which builds tick in every row?

A statement for every case

A statement that describes a pattern or property holding in all possible cases is a general statement. Arriving at it from examples is generalisation.

n and any multiple of n (such as 2n, 5n, 12n) always have HCF n, whatever n is. A single failing case would break it: m and m + 3 fail for m = 4 (4 and 7 have HCF 1).

A statement describing a pattern or property that holds in all possible cases is a general statement, and the process of arriving at it is generalisation. It can be described using algebra.

• Some cases | In letters
• 6 and 18 = 3 × 6 | n and 3n
• 7 and 35 = 5 × 7 | n and 5n
• 12 and 60 = 5 × 12 | n and 5n

Notes

A statement describing a pattern or property that holds in all possible cases is a general statement; arriving at it is generalisation. Algebra describes it: the HCF of n and any multiple of n is n.

Check yourself

Which of these is a general statement?

The first number is 7k. The other number is 3 × 7k = 21k. Take k = 2. What is the HCF of the two numbers?

Answer: 14

For k = 2 the numbers are 14 and 42. 42 = 3 × 14, so the HCF is 14, which is 7k.

Anil says: “The HCF of m and m + 3 is m.” He checks m = 3: the numbers are 3 and 6, and the HCF is 3. What is wrong with his claim?

If m is a number, which could be the other number so that the HCF of the pair is m for every m?

• The HCF of 6 and 18 is 6.. This is true, but it is about one pair only. It is a single case.
• The HCF of n and 5n is n, for every number n. — correct. Yes! The letter n stands for every number, so it describes all the cases.
• The HCF of 4 and 20 is 4.. This is one pair of numbers. A general statement must cover all cases.
• Nothing: it worked for m = 3.. It worked once. But for m = 4 the numbers are 4 and 7, with HCF 1, not 4.
• 3 is not a number we can use for m.. m can be any number. The trouble is the other cases, such as m = 4.
• One case does not make it general: for m = 4 the numbers are 4 and 7, and the HCF is 1. — correct. Yes! A claim for every m must hold for every m. One case where it fails (a counterexample) is enough.
• m + 9. For m = 4 this gives 4 and 13, whose HCF is 1, not 4.
• 9m — correct. Yes! 9m is a multiple of m, so m is the HCF whatever m is.
• m − 1. For m = 5 this gives 5 and 4, whose HCF is 1, not 5.