Online Test — Number Systems
10 Questions • 15 min • Chapter MCQ
15:00
Question 1 of 10
medium
According to Euclid's Division Lemma, for a = 87 and b = 13, what are q and r?
q=6, r=9
q=5, r=22
q=7, r=10
q=6, r=10
Explanation: 13×6=78, 87−78=9, and 0≤9<13 ✓
Question 2 of 10
medium
Which of the following is the prime factorization of 126?
2 × \(3^{2} \times 7\)
\(2^{2} \times 3 \times 7\)
2 × 3 × 21
2 × 7 × 9
Explanation: 126÷2=63, 63÷3=21, 21÷3=7 → 2 × \(3^{2} \times 7\)
Question 3 of 10
medium
Which of these numbers is irrational?
\(\sqrt{4}\)
\(\sqrt{9}\)
\(\sqrt{16}\)
\(\sqrt{2}\)
Explanation: \(\sqrt{4}\)=2, \(\sqrt{9}\)=3, \(\sqrt{16}\)=4 (all rational). \(\sqrt{2}\) cannot be written as p/q
Question 4 of 10
medium
The decimal expansion of 3/8 is:
Terminating
Non-terminating repeating
Non-terminating non-repeating
Cannot be determined
Explanation: 8=\(2^{3}\) → only prime factor 2, so terminating decimal
Question 5 of 10
medium
If a=17 and b=5 in Euclid’s Division Lemma, which equation is correct?
17=5×3+2
17=5×4+ (−3)
17=5×2+7
17=5×5+ (−8)
Explanation: 5×3=15, 15+2=17, remainder 2 < 5 ✓
Question 6 of 10
medium
What is \(0.\overline{63}\) as a fraction in simplest form?
7/11
63/100
21/33
6/9
Explanation: Let x=0.636363…; 100x=63.6363…; 99x=63; x=63/99=7/11
Question 7 of 10
medium
The Fundamental Theorem of Arithmetic guarantees:
Every number is prime
Prime factorization is unique
All numbers are composite
HCF and LCM are equal
Explanation: The theorem specifically states unique prime factorisation (order excepted)
Question 8 of 10
medium
Which of the following rational numbers has a terminating decimal expansion?
5/6
7/9
3/14
9/25
Explanation: 25=\(5^{2}\) → only prime 5, so terminating. Others have prime factors 3 or 7
Question 9 of 10
medium
Prove that 2\(\sqrt{5}\) is irrational. Which assumption starts the proof correctly?
Assume 2\(\sqrt{5}\) = p/q, p,q integers, q≠0, in simplest form
Assume \(\sqrt{5}\) is rational
Assume 2\(\sqrt{5}\) is rational with p=2
Assume 2\(\sqrt{5}\) is an integer
Explanation: The contradiction method starts by assuming the number equals p/q in lowest terms
Question 10 of 10
medium
A number has decimal expansion 0.12012001200012… This number is:
Rational and terminating
Rational and repeating
Irrational
An integer
Explanation: The pattern increases zeros each time, never repeats the same block cyclically