Shortcut for Divisibility by 9

Transcript

Shortcut for Divisibility by 9 A number is divisible if sum of digits is divisible by 9 Example: 1809 is divisible by 9 Sum of digits = 1 + 8 + 0 + 9 = 18 Since 18 is divisible by 9, ∴ 1809 is divisible by 9 666666 is divisible by 9 Sum of digits = 6 + 5 + 6 + 6 + 6 + 6 = 36 Since 36 is divisible by 9, ∴ 666666 is divisible by 9 Why do we check sum of digits to check divisibility by 9? Let’s do an example, we take the number 7309 Writing number as 7309 = 7 × 1000 + 3 × 100 + 9 × 1 Note that we can write 10 = 9 + 1 Where 9 is divisible by 9, and 1 is remainder Similarly, 100 = 99 + 1 1000 = 999 + 1 Here, 99, and 999 are divisible by 9, and 1 is the remainder Visualising 7309 = 7 × 1000 + 3 × 100 + 9 × 1 So, we write 7309 = 7 × 1000 + 3 × 100 + 9 × 1 = 7 × (999 + 1) + 3 × (99 + 1) + 9 × 1 = 7 × 999 + 7 × 1 + 3 × 99 + 3 × 1 + 9 × 1 = 7 × 999 + 3 × 99 + 7 + 3 + 9 Thus, we only check sum of digits for divisibility by 9 Checking the sum for divisible by 9 Divisible by 9