Ex 9.1, 16

Transcript

Ex 9.1, 16 Recall that a shortcut to check whether a given number is divisible by 3 is to add the digits of the number and check if the sum is a multiple of 3. Express the relationship between ‘a number is divisible by 3’ and ‘sum of the digits is a multiple of 3’ using ‘If-then’ sentences. The two statements are: Proposition: If a number is divisible by 3, then the sum of its digits is divisible by 3 Converse: If the sum of a number's digits is divisible by 3, then the number is divisible by 3 Both are true, by the divisibility rule for 3. Why does this rule work? Each place value is 1 more than a multiple of 3: 10=9+1 100=99+1 1000=999+1 and so on. For example, a three-digit number with digits 𝑎,𝑏,𝑐 is: 𝑵=𝟏𝟎𝟎𝒂+𝟏𝟎𝒃+𝒄 We can rewrite it as: 𝑵=𝟗𝟗𝒂+𝟗𝒃+(𝒂+𝒃+𝒄). Here, 99𝑎+9𝑏=3(33𝑎+3𝑏) is divisible by 3. Therefore:, if 𝒂+𝒃+𝒄 is divisible by 3, then 𝑁 is divisible by 3. Thus, we only need to check sum of digits divisible by 3 We can use the same argument works for any number of digits Each place value is 1 more than a multiple of 3: 10=9+1 100=99+1 1000=999+1 and so on. For example, a three-digit number with digits 𝑎,𝑏,𝑐 is: 𝑵=𝟏𝟎𝟎𝒂+𝟏𝟎𝒃+𝒄 We can rewrite it as: 𝑵=𝟗𝟗𝒂+𝟗𝒃+(𝒂+𝒃+𝒄). Here, 99𝑎+9𝑏=3(33𝑎+3𝑏) is divisible by 3. Therefore:, if 𝒂+𝒃+𝒄 is divisible by 3, then 𝑁 is divisible by 3. Thus, we only need to check sum of digits divisible by 3 We can use the same argument works for any number of digits