Which of the following numbers are divisible by both 3 and 9?
(i) 21303
(ii) 321
(iii) 290571
(iv) 4032

Which of the following numbers are divisible by 11?
(i) 06821
(ii) 725857
(iii) 2968152
(iv) 59862

To determine which numbers are divisible by both 3 and 9, and which numbers are divisible by 11, we can use some divisibility rules. Let's go through the steps:
Divisibility by 3 and 9:
A number is divisible by 3 if the sum of its digits is divisible by 3. Similarly, a number is divisible by 9 if the sum of its digits is divisible by 9.

(i) 21303

Sum of digits: 2 + 1 + 3 + 0 + 3 = 9
Since 9 is divisible by both 3 and 9, 21303 is divisible by both 3 and 9.


(ii) 321

Sum of digits: 3 + 2 + 1 = 6
6 is divisible by 3 but not by 9, so 321 is divisible by 3 but not by 9.


(iii) 290571

Sum of digits: 2 + 9 + 0 + 5 + 7 + 1 = 24
24 is divisible by 3 but not by 9, so 290571 is divisible by 3 but not by 9.


(iv) 4032

Sum of digits: 4 + 0 + 3 + 2 = 9
9 is divisible by both 3 and 9, so 4032 is divisible by both 3 and 9.



Divisibility by 11:
A number is divisible by 11 if the difference between the sum of the digits in odd positions and the sum of the digits in even positions is a multiple of 11 (including zero).

(i) 06821

Odd positions: 0 + 8 + 1 = 9
Even positions: 6 + 2 = 8
Difference: 9 − 8 = 1
Since 1 is not divisible by 11, 06821 is not divisible by 11.


(ii) 725857

Odd positions: 7 + 5 + 7 = 19
Even positions: 2 + 8 + 5 = 15
Difference: 19 − 15 = 4
Since 4 is not divisible by 11, 725857 is not divisible by 11.


(iii) 2968152

Odd positions: 2 + 6 + 1 + 2 = 11
Even positions: 9 + 8 + 5 = 22
Difference: 11 − 22 = − 11
Since − 11 is divisible by 11, 2968152 is divisible by 11.


(iv) 59862

Odd positions: 5 + 8 + 2 = 15
Even positions: 9 + 6 = 15
Difference: 15 − 15 = 0
Since 0 is divisible by 11, 59862 is divisible by 11.



In summary:

Numbers divisible by both 3 and 9: 21303 and 4032.
Numbers divisible by 11: 2968152 and 59862.

Answered by LiamAlexanderSmith | 2025-07-06

The numbers divisible by both 3 and 9 are 21303 and 4032. The numbers divisible by 11 are 2968152 and 59862.
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Answered by LiamAlexanderSmith | 2025-08-24