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Divisibility by 11

We know that a number is divisible by 11 if the difference
A 2-digit number is divisible by
between the sum of its digits in odd places and the sum of Note: 11 if its both digits are same,
the digits in even places is either 0 or a multiple of 11. otherwise not divisible by 11.
Since, the general form of a 3-digit number abc is,

abc = 100a + 10b + c = 99a + a + 11b – b + c = 11(9a + b) + (a – b + c)

Here, (a – b + c) = (a + c – b) is the difference of the sum of digits at odd places and the sum of digits
at even place.

So, abc is divisible by 11 if a + c – b is either 0 or a multiple of 11.
Similarly, abcd = 1000a + 100b + 10c + d = 1001a – a + 99b + b + 11c – c + d

= 11(91a + 9b + c) + (b + d) – (a + c)

where, (b + d) – (a + c) = sum of digits at odd places – sum of digits at even places of number abcd.
So, abcd is divisible by 11 if (b + d) – (a + c) is either 0 or a multiple of 11.

Quick Check

1. Which of the following numbers is divisible by 5 but not by 10?
(a) 210 (b) 125 (c) 500 (d) 755
2. Which of the following numbers is divisible by 3 and 9?
(a) 504 (b) 744 (c) 960 (d) 1224

Enrichment

Divisibility by 7 and 13
Divisibility by 7
A number is divisible by 7 if we take the last digit of the number, double it and subtract it from the rest of the
number (truncated number), the answer is either zero (0) or divisible by 7. Repeat the process if required.
For example, 665 is divisible by 7, as 66 – 5 × 2 = 56 ÷ 7 = 8 (Divisible by 7).
Thus we can say that, the number abc = 100a + 10b + c is divisible by 7, if {10(10a + b) – 2c)} is either zero (0)
or divisible by 7.
1. Check the divisibility of the following numbers by 7.
(a) 2394 (b) 87537

Divisibility by 13
If we multiply the last digit of the given number with 4 and add it to the rest truncated number. If the outcome
is divisible by 13, then the number is also divisible by 13. Repeat the process if required.
For example, 2795 is divisible by 13, as 279 + (5 × 4) = 299 ⇒ 29 + (9 × 4) = 65 ÷ 13 = 5 (Divisible by 13).
Thus we can say that, the number abc (100a + 10b + c) is divisible by 13, if {10(10a + b) + 4c)} is divisible by 13.

2. Check the divisibility of the following numbers by 13.
(a) 1443 (b) 5626

349 Playing With Numbers