MHB Solving Multiples Problems with Distinct Digits

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The discussion focuses on finding the number of four-digit multiples of 99 with distinct digits. It establishes that there are a total of 91 four-digit multiples of 99. Through algebraic expressions, it identifies that 9 multiples arise from the first equation (12 + 11x) and 10 from the second (19 + 9y), with one exceptional case. After accounting for these, the final count of distinct digit multiples is determined to be 71. The thread highlights patterns in the generated multiples, particularly noting the occurrence of repeated digits and palindromic numbers.
Marcelo Arevalo
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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?
 
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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.
 
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..
 
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