Question

Find the sum of all four - digit numbers divisible by 11 formed by using the digits 0 and 5 repeating any number of times in the number .
Options
i) 16060 ii) 21210
iii) 56760 iv) 27720

Answer 1

The sum of all four-digit numbers divisible by 11, formed by using the digits 0 and 5 repeating any number of times in the number, is 16060. Hence, option (i) is correct.

• Given that all the four digit numbers should be comprising of 0 and 5 only.

Now, a number cannot start from zero, or else, it would be considered as a three digit number only.

• Starting with 5, the possible four-digit numbers are :

(i) When 5 is used only once :

5000

(ii) When 5 is used twice :

5500, 5050, 5005

(iii) When 5 is used thrice :

5550, 5505, 5055

(iv) When 5 is used four times :

5555

• Now, a number is said to be divisible by 11 if on adding the digits in the odd positions (1st and 3rd for a 4-digit number) and the digits in the even positions (2nd and 4th for a 4-digit number) individually, and then subtracting the respective sums from each other, the difference results to 0 or 11.

• Let us consider all the numbers formed one by one :

(i) 5000

5 + 0 = 5

0 + 0 = 0

5 - 0 = 5 ( Not divisible by 11)

(ii) (a) 5500

5 + 0 = 5

5 + 0 = 5

5 - 5 = 0 (Divisible by 11)

(b) 5050

5 + 5 = 10

0 + 0 = 0

10 - 0 = 10 (Not divisible by 11)

(c) 5005

5 + 0 = 5

0 + 5 = 5

5 - 5 = 0 ( Divisible by 11)

(iii) (a) 5550

5 + 5 = 10

5 + 0 = 5

10 - 5 = 5 (Not divisible by 11)

(b) 5505

5 + 0 = 5

5 - 5 = 0

5 - 0 = 5 ( Not divisible by 11)

(c) 5055

5 + 5 = 10

0 + 5 = 5

10 - 5 = 5 (Not divisible by 11)

(iv) 5555

5 + 5 = 10

5 + 5 = 10

10 - 10 = 0 (Divisible by 11)

• Therefore, the numbers divisible by 11 are 5500, 5005, and 5555.

• Now, sum of 5500, 5005, and 5555 = 5500 + 5005 + 5555 = 16060

Answer 2

by 11 formed by using the digits 0 and 5 repeating any number of times in the number .

Options

i) 16060 ii) 21210

iii) 56760 iv) 27720