Base 7 representation of 2748 – UGC NET Question

Converting 2748 to base 7 requires repeated division by 7 and recording remainders. Performing this process yields digits that, read from last remainder to first, give 11004₇. We can verify by expanding 11004₇ as 1×7⁴ + 1×7³ + 0×7² + 0×7¹ + 4×7⁰, which equals 2401 + 343 + 0 + 0 + 4 = 2748. Hence 11004₇ is the correct base-7 representation, making option D correct.

Option A:
Option A has a similar-looking pattern of digits but a different combination of powers of 7. When expanded, it produces a decimal value slightly different from 2748. Therefore, it cannot represent 2748 exactly in base 7.

Option B:
Option B uses a larger digit in one of the higher positions, giving a decimal value greater than 2748 when converted. Since the expansion does not match 2748, this base-7 number is not correct.

Option C:
Option C uses a different arrangement of digits, yielding a smaller decimal value than 2748. Its place value expansion does not sum to 2748, so it is not an equivalent representation.

Option D:
Option D expands precisely to 2748 using the weights 7⁴, 7³ and 7⁰ with digits 1, 1 and 4 respectively and zeros in the remaining places. This correspondence between expansion and given decimal number ensures that 11004₇ is the only correct representation among the options.