- LPU MCA Semester 1
- PEA515
- Unit 1
PEA515 · Unit 1: Number system and averages
Number of factors, HCF and LCM
30 practice questions · in the MTE and ETE syllabus
Key points
- Factors count = product of (exponent + 1)
- HCF x LCM = product of the two numbers
- HCF for splitting and cutting, LCM for meeting again
- The product rule does not extend to three numbers
Write N as a product of prime powers, N = p^a x q^b x r^c. Then the number of factors is (a+1)(b+1)(c+1). For 360 = 2^3 x 3^2 x 5, that is 4 x 3 x 2 = 24. The sum of factors is the product of (p^(a+1) - 1)/(p - 1) over the primes.
HCF takes the smallest power of each common prime; LCM takes the largest power of every prime that appears. For two numbers, HCF x LCM = product of the numbers.
For fractions: HCF = HCF of numerators / LCM of denominators, and LCM = LCM of numerators / HCF of denominators (fractions in lowest terms).
Word problems: the largest tile or container size that divides lengths exactly uses HCF; the first time events coincide again, or the smallest number leaving a given remainder, uses LCM. If a number leaves remainder r on dividing by several numbers, it is LCM x k + r.
Exam trap: the product rule HCF x LCM = product holds for two numbers only, not three.
Try two questions
Two lights signal every 10 and 13 minutes. After how many minutes do they next signal together after starting together?
- 130
- 65
- 23
- 260
Show the answer
A. Next simultaneous time is the LCM of the intervals: 130 minutes.
Why not B: Divides the LCM by the number of bells.
Why not C: Adds the intervals.
Why not D: Doubles the LCM by counting two coincidences.
Find the number of divisors of 180.
- 5
- 16
- 8
- 18
Show the answer
D. Write 180 in prime powers and multiply (exponent + 1) terms: 18.
Why not A: Counts prime factors with repetition instead of using exponent plus one.
Why not B: Removes both 1 and the number although all factors were requested.
Why not C: Adds exponents instead of multiplying exponent-plus-one terms.