Divisibility and multiple test proofs

Understanding Divisibility and Multiple Test Proofs

• Divisibility refers to the ability of a number to divide another number without leaving a remainder. It is a key concept to understand for the topic of number theory.

• A multiple is a product of any quantity and an integer. In other words, if number ‘a’ can be expressed as ‘b x n’, where ‘n’ is an integer, ‘a’ is said to be a multiple of ‘b’.

The Concept of Test Proofs

• Test proofs are methods used to determine whether a number is divisible by another number. These are simple techniques that can often be performed mentally.

• A good example of this is the ‘divisible by 3’ test. We can determine whether a number is divisible by 3 by summing its digits and checking if the final sum is divisible by 3.

Divisibility Tests and Proofs

• Many divisibility tests are simple to understand and perform, but often the reasoning behind them is not as straightforward. A genuine understanding requires knowledge about the properties of numbers and basic algebra.

• For example, for the ‘divisible by 3’ test proof, it involves understanding that any integer can be expressed as a sum of the multiples of powers of 10, each multiplied by a digit. For example, 123 can be expressed as 1100 + 210 + 3.

• As 10 ≡ 1 (mod 3), any power of 10 will also be ≡ 1 (mod 3). This means we can reduce our multiple of power of 10 to just the digit (in the context of divisibility by 3). Therefore, our original number is divisible by 3 if and only if the sum of its digits is divisible by 3.

Multiple Test Proofs

• The logic behind multiple tests lies in understanding the factor structure of numbers. For example, if a number is a multiple of both 2 and 3, then it must also be a multiple of 6.

• This reasoning is based on prime factorisation and the structure of multiples. As 2 and 3 are prime numbers, a multiple of both will include both of these numbers in its prime factorization, making it also a multiple of 6.

Key Points to Remember

• Divisibility and multiple test proofs deepen your understanding of number theory and enhance your problem-solving skills. They also prepare you for the more complex topics involving modular arithmetic and congruences.

• Repeated practice with these tests and proofs will help you to consolidate your knowledge of numbers and build an intuitive understanding of their properties. Regular practice will also improve your mental arithmetic skills and your ability to quickly identify factors and multiples.