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Amy wrote an 5 digit number. That is divisible by 6 and 9. She spilled ink on the digit in the tens and the one's place by mistake. Write all such numbers possible in increasing order, the 10thousands place is 10,000 the thousands place is 2,000 and hundreds place is 400



Answer :

mhanifa

{12402, 12420, 12438, 12456, 12474, 12492}

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To solve the problem, let's understand the conditions and constraints:

  • We have a 5-digit number with specific digits.
  • The tens and ones digits are unknown.
  • The number should be divisible by both 6 and 9.


Given:

  • Ten thousand's place = 10000
  • Thousand's place = 2000
  • Hundred's place = 400
  • Tens and one's places are unknown.


The number can be represented as (10000 + 2000 + 400 + 10a + b), where (a) is the digit in the tens place and (b) is the digit in the one's place. This simplifies the number to:

  • 12400 + 10a + b

Conditions for divisibility:

  • A number is divisible by 6 if it is divisible by both 2 and 3.
  • A number is divisible by 9 if the sum of its digits is divisible by 9.

Given these conditions, we must find the values of (a) and (b) such that the number (12400 + 10a + b) satisfies both conditions. We list the resulting numbers in increasing order.


1. 12402 (a=0, b=2)

  • Divisible by 2 (last digit is even)
  • Sum of digits: (1 + 2 + 4 + 0 + 2 = 9) (divisible by 3 and 9)

2. 12420 (a=2, b=0)

  • Divisible by 2 (last digit is even)
  • Sum of digits: (1 + 2 + 4 + 2 + 0 = 9) (divisible by 3 and 9)

3. 12438 (a=3, b=8)

  • Divisible by 2 (last digit is even)
  • Sum of digits: (1 + 2 + 4 + 3 + 8 = 18) (divisible by 3 and 9)

4. 12456 (a=5, b=6)

  • Divisible by 2 (last digit is even)
  • Sum of digits: (1 + 2 + 4 + 5 + 6 = 18) (divisible by 3 and 9)

5. 12474 (a=7, b=4)

  • Divisible by 2 (last digit is even)
  • Sum of digits: (1 + 2 + 4 + 7 + 4 = 18) (divisible by 3 and 9)

6. 12492 (a=9, b=2)

  • Divisible by 2 (last digit is even)
  • Sum of digits: (1 + 2 + 4 + 9 + 2 = 18) (divisible by 3 and 9)


So, the numbers that Amy could have written, in increasing order, are:

  • {12402, 12420, 12438, 12456, 12474, 12492}