MHB Solving Multiples Problems with Distinct Digits

[Marcelo Arevalo]

How many 4 digit multiples of 99 are there whose digits are all distinct?

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99 x 11 = 1089
99 x 13 = 1287
99 x 14 = 1386
99 x 15 = 1485
and so on.. Is there any algebraic expressions for this?

 

[phymat]

I noticed a pattern while finding the multiples. If the number which is multiplied to 99 is 12+11x, or 19+9y where x can be any value from 0 to 8 and y can be any value from 0 to 9 respectively, then the multiple's digits are not distinct. Note that 101 is an exception. So, number of 4 digit multiples of 99 whose digits are all distinct are: $$91-9-10-1=71$$, where 91 is the total number of 4-digit multiples of 99, 9 is the number of 4-digit multiples of 99 which come through my first algebraic expression (12+11x), 10 is the number of 4-digit multiples of 99 which come through my second algebraic expression (19+9y), and 1 is the number of exceptional multiples of 99 which do not come through any of my above algebraic expressions.

 

[Marcelo Arevalo]

Using 1st equation
12 + 11x
12 + 11(0) = 12 * 99 = 1188
12 + 11(1) = 23 * 99 = 2277
12 + 11(2) = 34 * 99 = 3366
12 + 11(3) = 45 * 99 = 4455
12 + 11(4) = 56 * 99 = 5544
12 + 11(5) = 67 * 99 = 6633
12 + 11(6) = 78 * 99 = 7722
12 + 11(7) = 89 * 99 = 8811
12 + 11(8) = 100 * 99 = 9900

using equation 2:
19 + 9y
19 + 9(0) = 19 * 99 = 1881
19 + 9(1) = 28 * 99 = 2772
19 + 9(2) = 37 * 99 = 3663
19 + 9(3) = 46 * 99 = 4554
19 + 9(4) = 55 * 99 = 5445
19 + 9(5) = 64 * 99 = 6336
19 + 9(6) = 73 * 99 = 7227
19 + 9(7) = 82 * 99 = 8118
19 + 9(8) = 91 * 99 = 9009
19 + 9(9) = 100 * 99 = 9900

I noticed that in the first equation we get the double repeat digits. In the second we get the palindromic numbers.

all distinct digit.. are 71.

just want to clarify, 91 - 9 - 9 -1 = 72 .. I noticed the last number on equation 1 & equation 2 are the same..