4.5 Number System, LCM, HCF & Decimal Fractions

Classification of Numbers

There are 25 prime numbers below 100. Note that 1 is neither prime nor composite, a point that appears as an objective item on its own.

Co-prime numbers have HCF equal to 1; they need not themselves be prime, as 8 and 9 illustrate.

Divisibility Rules

Example. Is 4,53,728 divisible by 8? The last three digits are 728, and $728 = 8 \times 91$, so yes.

Example. Is 91,839 divisible by 11? Alternate sums are $9 + 8 + 9 = 26$ and $1 + 3 = 4$; wait, taking digits 9, 1, 8, 3, 9 from the left, odd positions give $9 + 8 + 9 = 26$ and even positions give $1 + 3 = 4$. The difference is 22, which is a multiple of 11, so yes.

HCF and LCM

The Highest Common Factor is the largest number dividing all the given numbers; the Lowest Common Multiple is the smallest number divisible by all of them. Both are read off the prime factorisation:

• HCF takes each common prime to its lowest power.
• LCM takes every prime appearing to its highest power.

Example. For $72 = 2^3 \times 3^2$ and $120 = 2^3 \times 3 \times 5$:

$$\text{HCF} = 2^3 \times 3 = 24, \qquad \text{LCM} = 2^3 \times 3^2 \times 5 = 360$$

The product relation

For exactly two numbers,

$$\text{HCF} \times \text{LCM} = \text{Product of the two numbers}$$

Check: $24 \times 360 = 8640 = 72 \times 120$. This relation does not extend to three or more numbers, which is a standard trap.

For fractions

$$\text{HCF of fractions} = \frac{\text{HCF of numerators}}{\text{LCM of denominators}}, \qquad \text{LCM of fractions} = \frac{\text{LCM of numerators}}{\text{HCF of denominators}}$$

Applications of HCF and LCM

Recognising which one a word problem needs is most of the work.

Worked example. Three conveyor drives at a coal handling plant complete a maintenance cycle every 12, 18 and 30 hours. If all three are serviced together now, when do they next coincide?

$$12 = 2^2\cdot3,\quad 18 = 2\cdot3^2,\quad 30 = 2\cdot3\cdot5 \;\Rightarrow\; \text{LCM} = 2^2\cdot3^2\cdot5 = 180 \text{ hours}$$

Worked example. Find the least number that leaves remainder 3 when divided by 8, 12 and 16. The LCM of 8, 12 and 16 is 48, so the answer is $48 + 3 = 51$.

Worked example. Find the greatest number that divides 43, 91 and 183 leaving the same remainder in each case. Take the differences: $91 - 43 = 48$, $183 - 91 = 92$, $183 - 43 = 140$. The HCF of 48, 92 and 140 is 4.

Remainders

For questions of the form "what is the remainder when $a^n$ is divided by $m$", use cyclicity. The units digit of powers repeats in a cycle of length at most 4:

Example. The units digit of $7^{83}$: the cycle length is 4, and $83 \div 4$ leaves remainder 3, so the units digit is the third in the cycle, which is 3.

Decimal Fractions

Terminating versus recurring

A fraction in lowest terms terminates as a decimal only if its denominator has no prime factor other than 2 and 5. So $3/8$ terminates because $8 = 2^3$, while $1/6$ does not because 6 contains the factor 3.

Converting recurring decimals

For a purely recurring decimal, the numerator is the repeating block and the denominator is as many nines as there are repeating digits:

$$0.\overline{36} = \frac{36}{99} = \frac{4}{11}$$

For a mixed recurring decimal, subtract the non-repeating part and use nines followed by zeros:

$$0.2\overline{45} = \frac{245 - 2}{990} = \frac{243}{990} = \frac{27}{110}$$

Useful fraction-to-percentage equivalents

Memorising these converts many percentage and ratio problems into single-step arithmetic:

With no negative marking on the CIL CBT, speed on this foundational topic frees time for the harder arithmetic later in Paper-I and for the engineering calculations in Paper-II.