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Algebra Arithmetic Math Olympiad PRMO

Digits of number | PRMO 2018 | Question 3

Try this beautiful problem from the Pre-RMO, 2018 based on Digits of number. You may use sequential hints to solve the problem.

Try this beautiful problem from the PRMO, 2018 based on Digits of number.

Digits of number – PRMO 2018


Consider all 6-digit numbers of the form abccba where b is odd. Determine the number of all such 6-digit numbers that are divisible by 7.

  • is 107
  • is 70
  • is 840
  • cannot be determined from the given information

Key Concepts


Algebra

Numbers

Multiples

Check the Answer


But try the problem first…

Answer: is 70.

Source
Suggested Reading

PRMO, 2018, Question 3

Higher Algebra by Hall and Knight

Try with Hints


First hint

abccba (b is odd)

=a(\(10^5\)+1)+b(\(10^4\)+10)+c(\(10^3\)+\(10^2\))

=a(1001-1)100+a+10b(1001)+(100)(11)c

=(7.11.13.100)a-99a+10b(7.11.13)+(98+2)(11)c

=7p+(c-a) where p is an integer

Second Hint

Now if c-a is a multiple of 7

c-a=7,0,-7

hence number of ordered pairs of (a,c) is 14

Final Step

since b is odd

number of such number=\(14 \times 5\)=70.

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