•If the HCF of two numbers is H, the numbers can be assumed as H × a and H × b such that a and b are coprime.

•While finding the largest/smallest number that leaves particular remainders when divided by certain numbers, do not forget to first eliminate options based on the remainder and divisor combinations, e.g. a number leaving a remainder of 5 on division by 8 has to be odd; a number leaving a remainder of 3 on division by 15 has to have the unit digit as 3 or 8; etc.

•LCM of multiples of 5 would end with just 0 or 5.

•HCF of a set of numbers will be odd if even one of the number in the given set is odd.

•Keep in mind- HCF will be lesser than or equal to the least of the numbers and LCM will be greater than or equal to the greatest of the numbers.

•LCM of fraction= LCM of numerators / HCF of denominators

•HCF of fraction= HCF of numerator / LCM of denominators

•ERROR PRONE AREA: While calculating HCF or LCM of fractions, the fractions should be in the most reducible form.

•The HCF of a set of numbers has to be a factor of the LCM of the set of numbers OR in other words, the LCM of a set of numbers must be a multiple of the HCF of the set of numbers. Thus if HCF of a set of numbers is a multiple of 5, the LCM of the set of numbers must also be a multiple of 5.

•Do not forget that for two numbers, the product of numbers = HCF × LCM

•If a series of numbers is of the type a × N + b, then consecutive numbers of this series differ by N. Conversely any series of numbers differing by N (basically an AP with common difference N) can be represented as a multiple of N ± x.

In any block of N consecutive numbers, there will be one and ONLY one number belonging to the above series. Thus from 1 to N, there exists one and ONLY one number belonging to the series. And so also from 101 to 100+N, or any such block of N numbers.
The above property can be used to identify the range of the smallest or largest n-digit number of the series. E.g. the smallest 4 digit number of the series 52N + 15 will be within 1000 and 1051 and the largest 3 digit number such number will lie within 948 and 999.

•Whenever the number of factors is the focus of any questions, train your mind to think in the following direction……to find the number of factors of any given number, factorize the number i.e. write it as powers of prime numbers 2a × 3b × 5c × 7d × 11e × ……Now the number of factors is (a + 1)(b + 1)(c + 1)(d + 1)(e + 1)……

•The number of ways a given number be written as a product of two numbers = ½ × the number of factor (if number of factors is even) OR
½ × (number of factors + 1) (if number of factors is odd). This will contain one way of writing the number as a product of similar numbers i.e. square of the square root of the given number.