Suppose today your friend is 12 years old and he was born in 1976.

[STEP 1] - You will arrive at the same number you started.

[STEP 2] - Now repeat this three-digit number to form a six-digit number:    362362

[STEP 3] - Dividing it by 7, we get:    51,766

[STEP 4] - Dividing 51,766 by 11 gives:    4701

[STEP 5] - Dividing 4701 by 13 eventually:    362

Suppose today your friend is 12 years old and he was born in 1976.

[STEP 1] - Consider the three-digit number: 789

[STEP 2] - Now repeat this three-digit number to form a six-digit number:    789789

[STEP 3] - Dividing it by 7, we get:    112,827

[STEP 4] - Dividing 112,827 by 11 gives:    10,257

[STEP 5] - Dividing 10,257 by 13 eventually:    789

[STEP 1] - Let us consider the number: 10

[STEP 2] - Doubling it, we get: 20

[STEP 3] - Now adding 5 and 12 successively, we get: 25 & 37

[STEP 4] - Subtracting 3 from the above yields: 34

[STEP 5] - Dividing the above number by 2 would make: 17

[STEP 6] - Subtracting 10 (which was the original number) from the above gives: 7

[STEP 1] - Let us consider any other number, say:    45

[STEP 2] - Doubling this number, we get:    90

[STEP 3] - Adding 5 to the above number and 12 successively, we get:    95 & 107

[STEP 4] - Subtracting 3 from above number gives:    104

[STEP 5] - Dividing the above number by 2 gives:    7

The answer is always 7. You can try the trick.

[STEP 1] - Let us consider the number as:    35

[STEP 2] - Multiplying this number by 3, it gives:    105

[STEP 3] - Adding 2 to the above number, we get:    107

[STEP 4] - Multiplying again by 3, we get:    321

[STEP 5] - Now adding a number, which is 2 more than the number first thought of (since 35 was the number initially thought of, we must add 37):    358

Now the number after the unit digit in the answer i.e. 35 remains the number initially meant.

[STEP 1] - Let us consider any other number, say: 7

[STEP 2] - Multiplying this number by 3 gives: 21

[STEP 3] - Adding 2 to the above number we get: 23

[STEP 4] - Multiplying the above number by 3 we get: 69

[STEP 5] - Now adding 9 to the above number (since it is two more than the original number): 78

You will find that the number after the unit digit in the answer is the number initially thought of i.e. 7.