Question: On which day in April is Gautam's birthday? Statements: Gautam was…

Concept: In a Data Sufficiency problem, a statement (alone, or together with another) is sufficient only when applying it yields exactly one possible value for what the question asks. If a statement leaves more than one possibility open, or cannot even be applied without an outside anchor, it is insufficient.

Application:

1. Statement II fixes the mother's own date of birth: she turns 55 years 4 months 5 days old on August 18 this year, so her birthday is August 18 minus 5 days minus 4 months minus 55 years = April 13. It does not, however, say anything about how Gautam's birthday relates to hers, so II alone cannot answer what is asked about Gautam.

2. Statement I fixes only the relationship between the two birthdays: Gautam was born exactly 28 years after his mother, meaning on the same calendar day and month as hers, 28 years later. It gives no absolute date on its own, so I alone cannot answer what is asked either.

3. Combining both: Statement II anchors the mother's birthday to April 13, and Statement I transfers that same day and month to Gautam. Together they fix Gautam's birthday as April 13 — one single, unambiguous date.

Cross-check: Using only I, countless day/month pairs are exactly 28 years apart, so no unique date follows. Using only II, the mother's birthday is fixed but nothing links it to Gautam, so his date remains unknown. Only when both statements are used together does every ambiguity disappear — confirming that both statements are needed jointly, and neither alone is enough.