Solving Multiples Problems with Distinct Digits

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  • Thread starter Marcelo Arevalo
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In summary, there are 71 4-digit multiples of 99 whose digits are all distinct. This is determined by the algebraic expressions 12 + 11x and 19 + 9y, where x can be any value from 0 to 8 and y can be any value from 0 to 9. However, 101 is an exception to this pattern. After considering all possibilities, the total number of multiples is 91, but after subtracting the 9 multiples from the first expression, the 10 multiples from the second expression, and the 1 exceptional multiple, we are left with 71 distinct multiples.
  • #1
Marcelo Arevalo
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How many 4 digit multiples of 99 are there whose digits are all distinct?

- - - Updated - - -

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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  • #2
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: \(\displaystyle 91-9-10-1=71\), 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.
 
  • #3
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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FAQ: Solving Multiples Problems with Distinct Digits

1. What are distinct digits in a multiple problem?

Distinct digits in a multiple problem refer to the unique numbers used in the calculation. For example, in the problem 12 x 3 = 36, the distinct digits are 1, 2, 3, and 6.

2. Why is it important to use distinct digits in multiple problems?

Using distinct digits ensures that each number is only used once in the calculation. This helps to avoid errors and ensures that the solution is accurate.

3. How can I identify if a multiple problem has distinct digits?

To identify if a multiple problem has distinct digits, you can list out all the numbers used in the calculation and check for any repetition. Alternatively, you can use each digit only once when solving the problem to ensure distinctness.

4. What are some strategies for solving multiple problems with distinct digits?

One strategy is to start by multiplying the largest numbers first, as this will limit the number of possibilities for the remaining digits. Another strategy is to use the distributive property to break down the problem into smaller parts.

5. Can I apply the same strategies for solving multiple problems with distinct digits to larger numbers?

Yes, the same strategies can be applied to larger numbers. However, it may be helpful to use a calculator or grid paper to keep track of the digits and their placement in the calculation.

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