UGC NET Questions (Paper – 1)

Converting 345 to base seven requires dividing by 7 and tracking remainders, which produces digits 1, 0, 0 and 2, giving 1002₇. Expanding 1002₇ as 1×7³ + 0×7² + 0×7¹ + 2×7⁰ gives 343 + 0 + 0 + 2 = 345. Hence, 1002 is the correct base-seven representation of 345.

Option A:
1003₇ expands as 1×343 + 0×49 + 0×7 + 3 = 343 + 0 + 0 + 3 = 346. This value is one more than 345, so this option is incorrect.

Option B:
1052₇ corresponds to 1×343 + 0×49 + 5×7 + 2 = 343 + 0 + 35 + 2 = 380. As this is significantly larger than 345, it cannot be the correct conversion.

Option C:
1002₇ represents 343 + 0 + 0 + 2 = 345, which exactly equals the decimal given. Its structure shows that only the highest and lowest positions carry nonzero digits in this base-seven form.

Option D:
1012₇ gives 1×343 + 0×49 + 1×7 + 2 = 343 + 0 + 7 + 2 = 352. Since 352 is larger than 345, this pattern is not the right representation.