Divisibles

Concatenate any 3-digit number with itself; e.g. for 835 it would be 835835. Divide it by 7.
The remainder will be zero!

Example: For 235, you would have 235235; where 235235 divided by 7 will be 33605 with 0 as the remainder.

Now, divide the quotient by 11. The remainder will be zero!

For the above example, 33605 divided by 11 will be 3055 with 0 as the remainder!

Again, divide the new quotient by 13. The remainder will be zero!

In the example, 3055 divided by 13 will be 235 and the remainder will be 0. Note that 235 is the original number of this example!

In summary:

235
235235 (concatenated by itself)
235235 / 7 = 33605 (remainder is zero)
33605 / 11 = 3055 (remainder is zero)
3055 / 13 = 235 (remainder is zero and 235 is the original number)

Another example; take 743 then:
743
743743 (concatenated by itself)
743743 / 7 = 106249 (remainder is zero)
106249 / 11 = 9659 (remainder is zero)
9659 / 13 = 743 (remainder is zero and 743 is the original number)

The reason is:

7 * 11 * 13 = 1001
and multiplying 1001 by any 3-digit number will be like concatenating it with itself; e.g. 100 * 1001 = 100100, or 532 * 1001 = 532532.

Hence, any number obtained by concatenating a 3-digit number with itself can be divided by 1001 and as 1001 is equal to 7 * 11 * 13, it can be divided by 7, 11, and 13.

By knowing the above fact, in a minute, out of the followings, find four numbers which are divisible by 13!

a) 713700
b) 845548
c) 145158
d) 987978
e) 426526
f) 469456
g) 363350

Answer:

a, c, f and g.
a) 713700 (as 713700 + 13 = 713713; 713713 / 13 = 54901, so:713700 / 13 = 54900)
c) 145158 (145158 – 13 = 145145)
f) 469456 (469456 + 13 = 469469)
g) 363350 (363350 + 13 = 363363)