Answered

3. Using divisibility tests, determine which of following numbers are divisible by 6:
(a) 297144
(b) 1258
(c) 4335
(f) 438750 (g) 1790184 (h) 12583
(d) 61233
(i) 639210
(e) 901352
(j) 17852
Using divisibility tests determine which of the following numbers are divisible by 11.



Answer :

GinnyAnswer
Alright, let's work through each step-by-step.

### Divisibility by 6
A number is divisible by 6 if and only if it is divisible by both 2 and 3.

#### Divisibility by 2:
A number is divisible by 2 if its last digit is even.

#### Divisibility by 3:
A number is divisible by 3 if the sum of its digits is divisible by 3.

Let's check each given number for divisibility by both 2 and 3.

(a) 297144
- Last digit is 4 (even) ➔ Divisible by 2
- Sum of digits = 2 + 9 + 7 + 1 + 4 + 4 = 27 (which is divisible by 3) ➔ Divisible by 3
- Therefore, 297144 is divisible by 6.

(b) 1258
- Last digit is 8 (even) ➔ Divisible by 2
- Sum of digits = 1 + 2 + 5 + 8 = 16 (which is not divisible by 3) ➔ Not divisible by 3
- Therefore, 1258 is not divisible by 6.

(c) 4335
- Last digit is 5 (odd) ➔ Not divisible by 2
- Therefore, 4335 is not divisible by 6.

(d) 61233
- Last digit is 3 (odd) ➔ Not divisible by 2
- Therefore, 61233 is not divisible by 6.

(e) 901352
- Last digit is 2 (even) ➔ Divisible by 2
- Sum of digits = 9 + 0 + 1 + 3 + 5 + 2 = 20 (which is not divisible by 3) ➔ Not divisible by 3
- Therefore, 901352 is not divisible by 6.

(f) 438750
- Last digit is 0 (even) ➔ Divisible by 2
- Sum of digits = 4 + 3 + 8 + 7 + 5 + 0 = 27 (which is divisible by 3) ➔ Divisible by 3
- Therefore, 438750 is divisible by 6.

(g) 1790184
- Last digit is 4 (even) ➔ Divisible by 2
- Sum of digits = 1 + 7 + 9 + 0 + 1 + 8 + 4 = 30 (which is divisible by 3) ➔ Divisible by 3
- Therefore, 1790184 is divisible by 6.

(h) 12583
- Last digit is 3 (odd) ➔ Not divisible by 2
- Therefore, 12583 is not divisible by 6.

(i) 639210
- Last digit is 0 (even) ➔ Divisible by 2
- Sum of digits = 6 + 3 + 9 + 2 + 1 + 0 = 21 (which is divisible by 3) ➔ Divisible by 3
- Therefore, 639210 is divisible by 6.

(j) 17852
- Last digit is 2 (even) ➔ Divisible by 2
- Sum of digits = 1 + 7 + 8 + 5 + 2 = 23 (which is not divisible by 3) ➔ Not divisible by 3
- Therefore, 17852 is not divisible by 6.

### Divisibility by 11
A number is divisible by 11 if the difference between the sum of the digits in the odd positions and the sum of the digits in the even positions is either 0 or divisible by 11.

Let's check each given number for divisibility by 11.

(a) 297144
- Odd positions: 2, 7, 4
- Even positions: 9, 1, 4
- Sum of odd positions = 2 + 7 + 4 = 13
- Sum of even positions = 9 + 1 + 4 = 14
- Difference = |13 - 14| = 1 ➔ Not divisible by 11

(b) 1258
- Odd positions: 1, 5
- Even positions: 2, 8
- Sum of odd positions = 1 + 5 = 6
- Sum of even positions = 2 + 8 = 10
- Difference = |6 - 10| = 4 ➔ Not divisible by 11

(c) 4335
- Odd positions: 4, 3
- Even positions: 3, 5
- Sum of odd positions = 4 + 3 = 7
- Sum of even positions = 3 + 5 = 8
- Difference = |7 - 8| = 1 ➔ Not divisible by 11

(d) 61233
- Odd positions: 6, 2, 3
- Even positions: 1, 3
- Sum of odd positions = 6 + 2 + 3 = 11
- Sum of even positions = 1 + 3 = 4
- Difference = |11 - 4| = 7 ➔ Not divisible by 11

(e) 901352
- Odd positions: 9, 1, 5
- Even positions: 0, 3, 2
- Sum of odd positions = 9 + 1 + 5 = 15
- Sum of even positions = 0 + 3 + 2 = 5
- Difference = |15 - 5| = 10 ➔ Not divisible by 11

(f) 438750
- Odd positions: 4, 8, 5
- Even positions: 3, 7, 0
- Sum of odd positions = 4 + 8 + 5 = 17
- Sum of even positions = 3 + 7 + 0 = 10
- Difference = |17 - 10| = 7 ➔ Not divisible by 11

(g) 1790184
- Odd positions: 1, 9, 1, 4
- Even positions: 7, 0, 8
- Sum of odd positions = 1 + 9 + 1 + 4 = 15
- Sum of even positions = 7 + 0 + 8 = 15
- Difference = |15 - 15| = 0 ➔ Divisible by 11

(h) 12583
- Odd positions: 1, 5, 3
- Even positions: 2, 8
- Sum of odd positions = 1 + 5 + 3 = 9
- Sum of even positions = 2 + 8 = 10
- Difference = |9 - 10| = 1 ➔ Not divisible by 11

(i) 639210
- Odd positions: 6, 9, 1
- Even positions: 3, 2, 0
- Sum of odd positions = 6 + 9 + 1 = 16
- Sum of even positions = 3 + 2 + 0 = 5
- Difference = |16 - 5| = 11 ➔ Divisible by 11

(j) 17852
- Odd positions: 1, 8, 2
- Even positions: 7, 5
- Sum of odd positions = 1 + 8 + 2 = 11
- Sum of even positions = 7 + 5 = 12
- Difference = |11 - 12| = 1 ➔ Not divisible by 11

### Summary:
- Divisible by 6: 297144, 438750, 1790184, 639210
- Divisible by 11: 1790184, 639210

Other Questions