Practice Questions for Algebra - Number System

1. Miscellaneous - 1 | Algebra - Number System

If (543)₆ = (317)_k, find k?

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2. Miscellaneous - 1 | Algebra - Number System

1331 is a perfect cube in which of the following bases?

(a)
3
(b)
7
(c)
6
(d)
More than 1 of the above

3. Miscellaneous - 1 | Algebra - Number System

(abcd)₁₃ – (dcba)₁₃ is always divisible by?

(a)
5
(b)
7
(c)
5 or 7
(d)
None of these

4. Miscellaneous - 1 | Algebra - Number System

A three-digit non-zero number (abc)₅ when converted to base 7 becomes (cba)₇. What is b?

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5. Miscellaneous - 1 | Algebra - Number System

(B)₁₂ + (BB)₁₂ + (BBB)₁₂ + (BBBB)₁₂ = (…………..)₁₀

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6. Miscellaneous - 1 | Algebra - Number System

If n = (111)₂ + (222)₃ + (333)₄ + (444)₅ + … + (FFF)₁₆. Find n?

(a)
${[\frac{16 (16 + 1)}{2}]}^{2}$
(b)
${[\frac{16 (16 + 1)}{2}]}^{2}$ - 15
(c)
${[\frac{16 (16 + 1)}{2}]}^{2}$ - 16
(d)
None of these

Answer: Option C

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Explanation :

(111)₂ = 1 × 2² + 1 × 2¹ + 1 = 2³– 1.

Similarly, (222)₃ = 3³– 1,

… and so on till

(FFF)16 = 16³ – 1.

∴ (111)₂ + (222)₃ + (333)₄ + (444)₅ + … + (FFF)₁₆ = (2³- 1) + (3³ - 1) + (4³- 1) + … + (16³- 1)

= (2³ + 2³ + … + 16³) - 15

= (1³ + 2³ + 2³ + … + 16³) - 16

${[\frac{16 (16 + 1)}{2}]}^{2}$ - 16

Hence, option (c).

7. Miscellaneous - 1 | Algebra - Number System

The number of three-digit numbers in septanary (base 7) system is?

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8. Miscellaneous - 1 | Algebra - Number System

Product of 44 and 11 represented when represented in a certain base is 3414. The number 3111 of this system, when converted to the decimal system is?

(a)
406
(b)
1086
(c)
213
(d)
691

9. Miscellaneous - 1 | Algebra - Number System

The cube root of (20311)₅ = (…………..)₁₀.

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10. Miscellaneous - 1 | Algebra - Number System

Is the number (27316)⁹ divisible by 8?

Type 1 if it is divisible.
Type 2 if it is not divisible.
Type 3 if it cannot be determined.

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Miscellaneous - 1 | Algebra - Number System

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