CBSE Class 8 Mathematics Factors and Multiples Notes Set 07

Factors and Multiples

If a number a divides b exactly, a is a factor of b and b is a multiple of a. The number 1 has exactly one factor, a prime number has exactly two factors (1 and itself), and 2 is the only even prime. Two numbers whose H.C.F. is 1 are called co-primes.
Divisibility tests: by 2 if the last digit is even; by 3 (or 9) if the digit sum is divisible by 3 (or 9); by 4 if the last two digits form a multiple of 4; by 5 if the last digit is 0 or 5; by 8 if the last three digits form a multiple of 8; by 11 if the difference between the sums of digits in odd and even places is 0 or a multiple of 11.

H.C.F. and L.C.M.

H.C.F.: the greatest number that divides all the given numbers. Prime factorisation method: multiply the common primes with their smallest powers. Division method: divide the larger by the smaller, then the divisor by the remainder, and so on; the last divisor is the H.C.F.
L.C.M.: the smallest number divisible by all the given numbers. Prime factorisation method: multiply all primes with their greatest powers. The H.C.F. is always a factor of the L.C.M.

Important Relations

For two numbers a and b: H.C.F. × L.C.M. = a × b. For co-primes the L.C.M. is the product. For fractions: H.C.F. = H.C.F. of numerators / L.C.M. of denominators, and L.C.M. = L.C.M. of numerators / H.C.F. of denominators (fractions in lowest terms).
For decimals, make the number of decimal places equal, remove the decimal points, find the H.C.F. or L.C.M., and mark the same number of places in the answer.

Word Problem Patterns

Greatest length that measures given lengths exactly, or greatest number that divides given numbers: H.C.F. Least number divisible by given numbers, or events repeating together: L.C.M.
Remainder r in every case: the number is (multiple of L.C.M.) + r for the least number problem, and the H.C.F. of (number − r) values for the greatest divisor problem. If the remainders differ but the difference between divisor and remainder is the same, subtract that difference from a multiple of the L.C.M.

Worked Examples

Example 1: H.C.F. of 36 and 60: 36 = 2² × 3², 60 = 2² × 3 × 5, so H.C.F. = 2² × 3 = 12 and L.C.M. = 2² × 3² × 5 = 180. Check: 12 × 180 = 2160 = 36 × 60.
Example 2: Least number divisible by 6, 8 and 12 that leaves remainder 5: L.C.M. = 24, so the number is 24 + 5 = 29.
Example 3: Two bells toll every 12 s and 18 s. They toll together after L.C.M. = 36 s.
Example 4: Greatest number dividing 70 and 105 exactly is H.C.F. = 35.