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Explore the Back-StoryTry this beautiful problem from the PRMO, 2018 based on Digits of number.

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

Algebra

Numbers

Multiples

Answer: is 70.

PRMO, 2018, Question 3

Higher Algebra by Hall and Knight

abccba (b is odd)

=a(+1)+b(+10)+c(+)

=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

Now if c-a is a multiple of 7

c-a=7,0,-7

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

since b is odd

number of such number==70.

- https://www.cheenta.com/smallest-perimeter-of-triangle-aime-2015-question-11/
- https://www.youtube.com/watch?v=ST58GTF95t4&t=140s

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

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

Algebra

Numbers

Multiples

Answer: is 70.

PRMO, 2018, Question 3

Higher Algebra by Hall and Knight

abccba (b is odd)

=a(+1)+b(+10)+c(+)

=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

Now if c-a is a multiple of 7

c-a=7,0,-7

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

since b is odd

number of such number==70.

- https://www.cheenta.com/smallest-perimeter-of-triangle-aime-2015-question-11/
- https://www.youtube.com/watch?v=ST58GTF95t4&t=140s

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