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Probability Level 4

Find the number of 6 digit numbers that can be formed using digits 5 \huge{5} and 7 \huge{7} such that they are d i v i s i b l e \color{#3D99F6} {divisible} by 35 \huge{35}

6 57 43 5 35 7 29

Tanishq Varshney by Tanishq Varshney , 23, India

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1 solution

Pop Wong
Jun 13, 2020

First we know the unit digit must 5. It remains 2 5 = 32 2^5=32 possible choices

Next, we check 555555 is divisible by 35 .

Except the unit digit, for each digit replaced by 7 the total difference is list below.

I used "1" to stand for the replacement of 5 by 7 of that digit.

1 0 n 10^n - digit replace 5 by 7 Mod(35) C1 | 35 C2 | 35 C3|35 C4|35
1 20 20 1 1 1
2 200 25 1 1
3 2000 5 1
4 20000 15 1 1
5 200000 10 1 1 1
total 35 35 35 70

So, there are 5 6-digit numbers formed by 5 or 7 which are divisible by 35:

  • 555555
  • 755755
  • 575575
  • 757575
  • 775775
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