Pariksha SaathiLesson 3 of 8 · Mathematics, Classes I-V (Paper I)
Multiplication and Division: Tables, Long Methods, Remainders, Factors, HCF and LCM
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What multiplication and division mean
Multiplication begins as counting equal groups (the Class 2 chapter Counting in Groups) and as repeated addition: 4 + 4 + 4 = 3 × 4 = 12. An array of 3 rows of 4 chairs, turned sideways, shows 4 × 3, which is why the order does not change the product. Division has two meanings: equal sharing (12 laddoos among 3 children, how many each?) and equal grouping (12 laddoos, 3 to a box, how many boxes?); both give 4. Grouping is repeated subtraction: 12 - 3 - 3 - 3 - 3 reaches 0 in 4 steps. Division is the inverse of multiplication: 3 × 4 = 12 gives 12 ÷ 3 = 4 and 12 ÷ 4 = 3.
Tables and their patterns
Tables are best built, not only recited. Products in the table of 5 end in 5 or 0, and the table of 4 doubles the table of 2. In the table of 9 the tens digit rises by 1 as the ones digit falls by 1, and the digits of 18, 27, 36 up to 90 add up to 9. A forgotten fact can be rebuilt: 6 × 7 = 5 × 7 + 7 = 42 and 9 × 8 = 10 × 8 - 8 = 72. Multiplying by 100 moves each digit two places left (47 × 100 = 4,700). To multiply a 2-digit number by 11, write the sum of its digits between them: 11 × 23 = 253, and carry when the sum is more than 9, as in 11 × 57 = 627.
Long multiplication
The written method rests on place value and the distributive property. For 236 × 47, multiply by 7 to get 1,652, then by 4 tens to get 9,440 (the zero shows that it is 40, not 4), and add: 11,092. The grid method splits both numbers by place value: (200 + 30 + 6) × (40 + 7) gives 8,000 + 1,400 + 1,200 + 210 + 240 + 42 = 11,092; the Class 5 chapter Ways to Multiply and Divide likewise shows children more than one way to multiply and to divide. Multiplication is commutative, associative (2 × 5 × 37 = 10 × 37 = 370) and distributive: 7 × 105 = 700 + 35 = 735 and 8 × 99 = 800 - 8 = 792. Doubling one number and halving the other keeps the product: 25 × 36 = 50 × 18 = 900.
Long division and remainders
Divide from the greatest place. In 5,346 ÷ 6, 5 thousands cannot be shared among 6, so 53 hundreds give 8 with 5 hundreds left, 54 tens give 9 and 6 ones give 1: the quotient is 891, and 891 × 6 = 5,346. In 2,437 ÷ 12, 24 hundreds give 2, 3 tens give 0 tens (a zero that must be written) and 37 ones give 3 with 1 left: quotient 203, remainder 1, since 12 × 203 + 1 = 2,437. The remainder is always less than the divisor, and in word problems its meaning decides the answer: 100 eggs packed 12 to a tray fill 8 trays with 4 left over, but 100 children need 9 vans of 12 seats. 0 divided by any non-zero number is 0, division by 0 has no meaning, and 12 ÷ 4 is not 4 ÷ 12.
Factors, multiples and primes
A factor divides a number exactly; a multiple is the number times 1, 2, 3 and so on. The Class 5 chapter Be My Multiple, I'll be Your Factor introduces both ideas through games, and factors can be found by making rectangles: 24 counters form 1 × 24, 2 × 12, 3 × 8 and 4 × 6 arrays, so 24 has 8 factors. Factors are limited, multiples never end, and every number is a factor and a multiple of itself. A prime has exactly 2 factors (2, 3, 5, 7, 11 ...), a composite number more than 2, and 1 is neither; 2 is the only even prime, and there are 25 primes below 100. Divisibility tests save long division: 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 number ends in 0 or 5. 7,236 (digit sum 18, ending in 36) is divisible by 2, 3, 4, 6 and 9 but not by 5.
HCF and LCM at primary level
The highest common factor (HCF) is the greatest number that divides both numbers; the lowest common multiple (LCM) is the smallest number that both divide. Listing is enough here: the factors of 12 are 1, 2, 3, 4, 6, 12 and of 18 are 1, 2, 3, 6, 9, 18, so the HCF is 6; the multiples of 12 are 12, 24, 36 ... and of 18 are 18, 36 ..., so the LCM is 36. For two numbers, HCF × LCM = their product: 6 × 36 = 216 = 12 × 18. Cutting into the largest equal pieces needs the HCF: ribbons of 24 cm and 36 cm give pieces of 12 cm. Happening together again needs the LCM: bells ringing every 4 and every 6 minutes ring together every 12 minutes, so after 9:00 a.m. they next ring together at 9:12 a.m.
Key facts
- Division has two meanings: sharing (12 among 3, how many each?) and grouping (12 in 3s, how many groups?); both give 4.
- Table of 9: the tens digit rises as the ones digit falls, and the digits of every product from 18 to 90 add up to 9.
- 236 × 47 = 1,652 + 9,440 = 11,092; the second line is 236 × 40, so it needs the zero.
- Distributive shortcuts: 7 × 105 = 700 + 35 = 735; 8 × 99 = 800 - 8 = 792; 11 × 57 = 627.
- 2,437 ÷ 12 = 203 remainder 1; check 12 × 203 + 1 = 2,437; the zero in 203 must not be dropped.
- Meaning of the remainder: 100 eggs, 12 to a tray, fill 8 trays with 4 left over; 100 children, 12 to a van, need 9 vans.
- 24 has 8 factors (1, 2, 3, 4, 6, 8, 12, 24); 1 is neither prime nor composite; 2 is the only even prime; there are 25 primes below 100.
- 7,236: digit sum 18 and last two digits 36, so it is divisible by 2, 3, 4, 6 and 9, but not by 5 or 10.
- HCF of 12 and 18 = 6, LCM = 36, and 6 × 36 = 216 = 12 × 18.
- Largest equal pieces means HCF (24 cm and 36 cm give 12 cm pieces); together again means LCM (every 4 and 6 minutes gives every 12 minutes).
Quick revision
Multiplication: equal groups, repeated addition, arrays; division: sharing or grouping, the inverse of multiplication; 236 × 47 = 11,092; 2,437 ÷ 12 = 203 remainder 1; remainder < divisor, round up or down as the situation needs; × is commutative, associative, distributive; ÷ 0 has no meaning; prime has exactly 2 factors, 1 is neither, 2 the only even prime; HCF for largest equal parts, LCM for together again; HCF × LCM = product.