Pariksha SaathiLesson 1 of 8 · Mathematics, Classes VI-VIII (Paper II)
Number System: Integers, Rational Numbers, Divisibility, HCF and LCM
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Whole numbers and integers
The natural numbers 1, 2, 3, ... become the whole numbers when 0 is added, and the integers when −1, −2, −3, ... are added too. Whole numbers are closed under addition and multiplication but not under subtraction (3 − 5) or division (5 ÷ 2); integers are closed under subtraction as well, but not under division, since (−9) ÷ 2 is not an integer. Multiplication distributes over addition, a × (b + c) = a × b + a × c, which gives quick products: 854 × 102 = 85,400 + 1,708 = 87,108. For any integer a, a ÷ (−1) = −a, 0 ÷ a = 0 when a ≠ 0, and a ÷ 0 is not defined.
Rational numbers
A rational number can be written as p/q, where p and q are integers and q ≠ 0, so every integer (−5 = −5/1) and every terminating decimal (0.75 = 3/4) is rational. In standard form the denominator is positive and the numerator and denominator have no common factor except 1: −45/30 and 36/−24 both become −3/2. To compare, use a common positive denominator: −3/4 = −9/12 and −5/6 = −10/12, so −3/4 > −5/6. Rational numbers are closed under addition, subtraction and multiplication, but not under division, as division by 0 is not defined. The additive inverse of a/b is −a/b and its reciprocal is b/a, so the reciprocal of −7/9 is −9/7, not 9/7, and 0 has no reciprocal.
Factors, multiples and primes
A prime has exactly two factors, a composite more than two, and 1 is neither. There are 25 primes up to 100, which the Sieve of Eratosthenes finds by striking out the multiples of 2, 3, 5 and 7 on a 100-chart; 2 is the only even prime, 4 the smallest composite and 9 the smallest odd composite. Twin primes differ by 2 (11 and 13); co-prime numbers have no common factor except 1 but need not be prime (8 and 15). A perfect number equals the sum of its other factors: 6 = 1 + 2 + 3 and 28 = 1 + 2 + 4 + 7 + 14.
Divisibility rules
Test 3 and 9 by the digit sum, 4 by the last two digits, 8 by the last three, 6 by both 2 and 3, and 11 by the difference between the sums of the digits in odd and even places, which must be 0 or a multiple of 11. For 61,809 the odd places give 9 + 8 + 6 = 23 and the even places 0 + 1 = 1; the difference 22 is a multiple of 11, and 61,809 = 11 × 5,619. If 21y5 is a multiple of 9, 8 + y must be 9 or 18, so the digit y is 1; if 31z5 is a multiple of 3, z is 0, 3, 6 or 9. General form explains the tricks: the two-digit number ab is 10a + b, so ab + ba = 11(a + b) and ab − ba = 9(a − b), and for three digits abc − cba = 99(a − c).
Prime factorisation, HCF and LCM
Every composite number is a product of primes in exactly one way apart from order, found by a factor tree or by repeated division. The HCF takes the common primes at their lowest powers and the LCM every prime at its highest. For 72 = 2³ × 3², 108 = 2² × 3³ and 180 = 2² × 3² × 5, the HCF is 2² × 3² = 36 and the LCM is 2³ × 3³ × 5 = 1,080. HCF × LCM = product holds for two numbers only: 36 × 216 = 7,776 = 72 × 108, but 36 × 1,080 is not 72 × 108 × 180.
Word problems
'Greatest' or 'longest that measures exactly' points to the HCF, 'least' or 'together again' to the LCM; a remainder is first subtracted from the numbers for the HCF but added to the LCM at the end. The greatest number dividing 245 and 1,029 with remainder 5 is the HCF of 240 and 1,024, that is 16. The least number leaving remainder 5 when divided by 6, 15 and 18 is their LCM plus 5: 90 + 5 = 95. The smallest three-digit number divisible by 6, 8 and 12 is the first three-digit multiple of their LCM 24, that is 120, and the greatest is 24 × 41 = 984.
In the classroom
Typical errors are calling 1 a prime, thinking co-prime numbers must be prime, taking the LCM to be the product (the LCM of 6 and 8 is 24, not 48), and assuming that a number divisible by 4 and 6 is divisible by 24 (12 is not; the rule needs co-prime divisors, as 2 and 9 for 18). The teacher lets pupils arrange 12 counters into every possible rectangle (1 × 12, 2 × 6, 3 × 4) to see factors in pairs, and marks multiples on a number line so that the first common mark shows the LCM.
Key facts
- Closure: whole numbers are not closed under subtraction (3 − 5 = −2), integers are not closed under division ((−9) ÷ 2 = −4.5), and rational numbers are closed under addition, subtraction and multiplication.
- Distributive law for quick multiplication: 854 × 102 = 854 × 100 + 854 × 2 = 85,400 + 1,708 = 87,108.
- Standard form of a rational number: −45/30 = −3/2; comparing, −3/4 = −9/12 is greater than −5/6 = −10/12.
- The reciprocal of −7/9 is −9/7 and its additive inverse is 7/9; 0 has no reciprocal.
- Divisibility by 11: for 61,809, (9 + 8 + 6) − (0 + 1) = 22, a multiple of 11, and 61,809 = 11 × 5,619.
- Primes: there are 25 primes up to 100; 2 is the only even prime; 1 is neither prime nor composite; 6 and 28 are perfect numbers.
- 72 = 2³ × 3², 108 = 2² × 3³ and 180 = 2² × 3² × 5, so their HCF is 36 and their LCM is 1,080.
- Remainders: the greatest number dividing 245 and 1,029 with remainder 5 is HCF(240, 1,024) = 16; the least number leaving remainder 5 on division by 6, 15 and 18 is 90 + 5 = 95.
- General form: ab + ba = 11(a + b), so 47 + 74 = 121 = 11 × 11; ab − ba = 9(a − b), so 74 − 47 = 27 = 9 × 3.
Quick revision
Natural numbers are whole numbers, whole numbers are integers, integers are rational; whole numbers not closed under subtraction, integers not under division; rational p/q with q ≠ 0, standard form has a positive denominator and no common factor; reciprocal of −7/9 is −9/7; 25 primes up to 100, 1 neither prime nor composite; rule for 11: alternate digit sums differ by 0 or a multiple of 11; HCF lowest powers, LCM highest; HCF × LCM = product for two numbers only; with remainder r use HCF of (numbers − r) or LCM + r.