4
Divisibility tests for 9 and 3
Textbook, pp. 13–15
GoalKnow the divisibility tests for 9 and 3; decide divisibility from the sum of digits.
New words
sum of digits · raqamlar yig‘indisidivisibility by 9 · 9 ga bo‘linishdivisibility by 3 · 3 ga bo‘linishtest / sign · belgi
Explanation
If the sum of a number's digits is divisible by 9, the number is divisible by 9. If the sum of its digits is divisible by 3, the number is divisible by 3. For example, in 4,527 we get 4 + 5 + 2 + 7 = 18, and 18 is divisible by both 9 and 3, so 4,527 is too. Every number divisible by 9 is also divisible by 3, but not the other way round: 12 is divisible by 3 but not by 9. If the digit sum is not divisible, the number is not divisible either.
Worked examples
2,736: 2 + 7 + 3 + 6 = 18, so 2,736 is divisible by 9 and by 3.
1,245: 1 + 2 + 4 + 5 = 12. 12 is divisible by 3 but not by 9, so 1,245 is divisible by 3 but not by 9.
Class activity
“Digit-sum relay”: two teams line up at the board. The teacher says a number, the first pupil writes the digit sum, the second writes “yes/no” for divisibility. The team that gets everything right wins.
Practice
1
Is 4,527 divisible by 9?
Yes, the digit sum is 18, which is divisible by 9.
2
Find the sum of digits of 7,308.
18
3
Which digits can replace * so that 5*4 is divisible by 3?
0, 3, 6, 9
4
12 is divisible by 3 but not by 9. Does this contradict “a number divisible by 9 is always divisible by 3”?
No. The statement goes from 9 to 3, not from 3 to 9. Since 9 = 3 · 3, divisibility by 9 gives divisibility by 3.