How many pairs of digits are there in the number 6281837, each of which has as…
2025
How many pairs of digits are there in the number 6281837, each of which has as many digits between them (in both the forward and backward direction) as they have between them in the natural number series?
- A.
Two
- B.
More than four
- C.
Four
- D.
One
- E.
Three
Show answer & explanation
Correct answer: E
Concept
A qualifying pair of digits is one where the count of digits lying strictly between the two digits in the given number equals the count of whole numbers lying strictly between those two digit values in the natural number series (… 1, 2, 3, 4 …). The position-gap is checked in both directions, so the order in which the two digits appear does not matter — only the size of the gap does.
Application
Write the number with positions: 6(1), 2(2), 8(3), 1(4), 8(5), 3(6), 7(7). For every pair, compare the positional gap (digits sitting between them) with the value gap (numbers sitting between them in 1–9):
Digit pair | Gap in the number | Gap in the number series |
|---|---|---|
6 and 8 | 1 digit between (the 2) | 1 number between (the 7) |
2 and 7 | 4 digits between (8, 1, 8, 3) | 4 numbers between (3, 4, 5, 6) |
1 and 3 | 1 digit between (the 8) | 1 number between (the 2) |
Each of these three pairs has equal gaps, so each one qualifies.
Cross-check
All other pairs fail. For instance, 8 and 1 sit two digits apart in the number but seven numbers apart in the series; 6 and 2 sit zero digits apart but five numbers apart. Scanning every remaining pair the same way leaves exactly three matches.
Hence the number of qualifying pairs is 3.