Find the odd number pair from the given alternatives.
2017
Find the odd number pair from the given alternatives.
- A.
46 – 10
- B.
42 – 33
- C.
25 – 43
- D.
12 – 91
Attempted by 103 students.
Show answer & explanation
Correct answer: D
A number's digit sum, reduced repeatedly until a single digit remains (its digital root), gives the number's remainder on division by 9. Two numbers share the same digital root exactly when their difference is a multiple of 9. Each option here is a number pair; three pairs share a digital root and one does not.
46 - 10: digit sum of 46 is 4 + 6 = 10, which reduces further to 1 + 0 = 1 (digital root 1); digit sum of 10 is 1 + 0 = 1 (digital root 1). Same digital root, so this pair is consistent.
42 - 33: digit sum of 42 is 4 + 2 = 6 (digital root 6); digit sum of 33 is 3 + 3 = 6 (digital root 6). Same digital root, so this pair is consistent.
25 - 43: digit sum of 25 is 2 + 5 = 7 (digital root 7); digit sum of 43 is 4 + 3 = 7 (digital root 7). Same digital root, so this pair is consistent.
12 - 91: digit sum of 12 is 1 + 2 = 3 (digital root 3); digit sum of 91 is 9 + 1 = 10, which reduces further to 1 + 0 = 1 (digital root 1). The digital roots 3 and 1 do not match, so this pair breaks the pattern.
Cross-check with differences: a matched digital-root pair always differs by a multiple of 9 - 46 minus 10 = 36 = 9 times 4, 42 minus 33 = 9 = 9 times 1, and 25 minus 43 = -18, a multiple of 9. But 12 minus 91 = -79, and 79 is not a multiple of 9, confirming the mismatch found above.
12 - 91 is the only pair whose numbers do not share a digital root (equivalently, the only pair whose difference is not a multiple of 9), so it is the odd number pair.