A lucky number is a positive integer which is 19 times the sum of its digits...??

Question

how many different lucky numbers are there?

Answer 1

The above answers are wrong. The first one is just laughing at you childishly. The second provides no explanation and includes 0 (0 is not positive). The third is confused about your definition of lucky number.

Call the digits a(i) and the number a.

19Σ a(i) = Σ10^i a(i) = a

It is clear then that our numbers have to have to be less than 570, since if it were more than 570, if it has 3 digits:

19Σa(i) ≤ 3×19×10 < 570 ≤ a

And if it has n>3 digits:

19Σa(i) ≤ n×19×10 < 10^n ≤ a

What if it had two digits?

19(a+b) = 10a + b ??
Impossible!!
19a + 19b > 10a + b

So it must be between 100 and 570.

In the 100s:
19(1+a+b) = 100 + 10a + b
19 + 19a + 19b = 100 + 10a + b
9a + 18b = 81
a + 2b = 9

Notice that a must be odd.
a = 1, b = 4, lucky number = 114
a = 3, b = 3, lucky number = 133
a = 5, b = 2, lucky number = 152
a = 7, b = 1, lucky number = 171
a = 9, b = 0, lucky number = 190

In the 200s:
19(2+a+b) = 200 + 10a + b
38 + 19a + 19b = 200 + 10a + b
9a + 18b = 168
a + 2b = 18
a has to be even here.
a = 0, b = 9 → 209
a = 2, b = 8 → 228
a = 4, b = 7 → 247
a = 6, b = 6 → 266
a = 8, b = 5 → 285

In the 300s:
19(3+a+b) = 300 + 10a + b
57 + 19a + 19b = 300 + 10a + b
9a + 18b = 243
a + 2b = 27
Here a has to be odd again:
a = 1, b = 13 → impossible (b is a digit, b ≠ 13)
a = 3, b = 12 → impossible
a = 5, b = 11 → impossible
a = 7, b = 10 → impossible
a = 9, b = 9 → 399

In the 400s:
19(4+a+b) = 400 + 10a + b
76 + 19a + 19b = 400 + 10a + b
9a + 18b = 324
a + 2b = 36
This is always impossible, since:
a+2b < 9+2(9) = 27 < 36

In the 500s:
19(5+a+b) = 500 + 10a + b
95 + 19a + 19b = 500 + 10a + b
9a + 18b = 405
a + 2b = 45
Also impossible, since:
a+2b < 9+2(9) = 27 < 45

114, 133, 152, 171, 190, 209, 228, 247, 266, 285, 399 are the only ones. That's 11 total.

Answer 2

If ab is the lucky number, what you mean is it is 19 (a + b). Example:

19 itself since the sum of the digits is 1 + 9 = 10 = 1 and 19 is 19 times 1.

38 is another number as per that formula since 3 + 8 = 11 = 2

Similarly all multiples of 19 like 57, 76, 95, 114, 133, 152, 171 meet the criteria.

190 and above don't meet the norm.

Answer 3

By your definition there are twelve lucky numbers:

0
114
133
152
171
190
209
228
247
266
285
399

Answer 4

Lmfao @ you. Or maybe not. Please explain, in detail, luck and/or lucky numbers.