If a three-digit number abc minus cba is a positive multiple of 7, how many distinct numbers can fulfill this condition?

To determine how many distinct three-digit numbers ( abc ) fulfill the condition that ( abc - cba ) is a positive multiple of 7, we first need to express ( abc ) and ( cba ) in terms of their digits.

Let:

( abc = 100a + 10b + c )

( cba = 100c + 10b + a )

Now, we can subtract ( cba ) from ( abc ):

[

abc - cba = (100a + 10b + c) - (100c + 10b + a)

]

[

= 100a + c - 100c - a

]

[

= 99a - 99c

]

[

= 99(a - c)

]

Next, we need to ensure that ( 99(a - c) ) is a positive multiple of 7. Since 99 is not a multiple of 7, we need ( (a - c) ) itself to be a positive multiple of 7.

The values for ( a ) and ( c ) can range from 0 to 9, but since (