Worksheet focused on HCF and LCM concepts, including questions on co-prime numbers, factorization, and methods to find highest common factor and lowest common multiple.

I. Fill in the blanks:

(i) The HCF of two numbers is 1, they are called co-prime numbers.
→ Because if their highest common factor is only 1, they share no other common factors — that’s what “co-prime” means.

(ii) The LCM of two or more numbers cannot be smaller than any one of the numbers.
→ LCM is a multiple — so it must be at least as big as the biggest number.

(iii) The HCF of the given numbers cannot be larger than the number themselves.
→ HCF is a divisor — so it can’t be bigger than the smallest number.

(iv) The HCF of 7 and 35 is 7.
→ 7 divides both 7 and 35 (35 ÷ 7 = 5), and there’s no bigger number that does.

(v) The LCM of 7 and 35 is 35.
→ 35 is a multiple of both 7 and 35 — and it’s the smallest such number.

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II. Find highest common factor by complete factorisation:

(i) 48, 56, 72
Break each into prime factors:
48 = 2⁴ × 3
56 = 2³ × 7
72 = 2³ × 3²
Common prime factor: 2³ = 8 → HCF = 8

(ii) 198, 360
198 = 2 × 3² × 11
360 = 2³ × 3² × 5
Common: 2 × 3² = 2 × 9 = 18 → HCF = 18

(iii) 102, 68, 136
102 = 2 × 3 × 17
68 = 2² × 17
136 = 2³ × 17
Common: 2 × 17 = 34 → HCF = 34

(iv) 1024, 576
1024 = 2¹⁰
576 = 2⁶ × 3²
Common: 2⁶ = 64 → HCF = 64

(v) 405, 783, 513
405 = 3⁴ × 5
783 = 3³ × 29
513 = 3³ × 19
Common: 3³ = 27 → HCF = 27

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III. Find H.C.F. by long division method:

(i) 84, 144
Divide 144 by 84 → remainder 60
Divide 84 by 60 → remainder 24
Divide 60 by 24 → remainder 12
Divide 24 by 12 → remainder 0 → HCF = 12

(ii) 120, 168
168 ÷ 120 = 1 rem 48
120 ÷ 48 = 2 rem 24
48 ÷ 24 = 2 rem 0 → HCF = 24

(iii) 430, 516, 817
First find HCF of 430 and 516:
516 ÷ 430 = 1 rem 86
430 ÷ 86 = 5 rem 0 → HCF(430,516) = 86
Now HCF of 86 and 817:
817 ÷ 86 = 9 rem 43 (since 86×9=774, 817-774=43)
86 ÷ 43 = 2 rem 0 → HCF = 43

(iv) 632, 790, 869
HCF(632,790):
790 ÷ 632 = 1 rem 158
632 ÷ 158 = 4 rem 0 → HCF = 158
Now HCF(158,869):
869 ÷ 158 = 5 rem 79 (158×5=790, 869-790=79)
158 ÷ 79 = 2 rem 0 → HCF = 79

(v) 291, 582, 776
Note: 582 = 2 × 291 → HCF(291,582)=291
Now HCF(291,776):
776 ÷ 291 = 2 rem 194 (291×2=582, 776-582=194)
291 ÷ 194 = 1 rem 97
194 ÷ 97 = 2 rem 0 → HCF = 97

(vi) 219, 1321, 2320, 8526
Start with 219 and 1321:
1321 ÷ 219 = 6 rem 7 (219×6=1314, 1321-1314=7)
219 ÷ 7 = 31 rem 2
7 ÷ 2 = 3 rem 1
2 ÷ 1 = 2 rem 0 → HCF(219,1321)=1
Since HCF is already 1, adding more numbers won’t change it → HCF = 1

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IV. Find lowest common multiple:

(i) 16, 24, 40
Prime factors:
16 = 2⁴
24 = 2³ × 3
40 = 2³ × 5
LCM = 2⁴ × 3 × 5 = 16 × 3 × 5 = 240

(ii) 40, 56, 60
40 = 2³ × 5
56 = 2³ × 7
60 = 2² × 3 × 5
LCM = 2³ × 3 × 5 × 7 = 8 × 3 × 5 × 7 = 840

(iii) 207, 138
207 = 3² × 23
138 = 2 × 3 × 23
LCM = 2 × 3² × 23 = 2 × 9 × 23 = 414

(iv) 72, 96, 120
72 = 2³ × 3²
96 = 2⁵ × 3
120 = 2³ × 3 × 5
LCM = 2⁵ × 3² × 5 = 32 × 9 × 5 = 1440

(v) 120, 150, 135
120 = 2³ × 3 × 5
150 = 2 × 3 × 5²
135 = 3³ × 5
LCM = 2³ × 3³ × 5² = 8 × 27 × 25 = 5400

(vi) 102, 170, 136
102 = 2 × 3 × 17
170 = 2 × 5 × 17
136 = 2³ × 17
LCM = 2³ × 3 × 5 × 17 = 8 × 3 × 5 × 17 = 2040

Final Answer:
I. (i) co-prime, (ii) smaller, (iii) larger, (iv) 7, (v) 35
II. (i) 8, (ii) 18, (iii) 34, (iv) 64, (v) 27
III. (i) 12, (ii) 24, (iii) 43, (iv) 79, (v) 97, (vi) 1
IV. (i) 240, (ii) 840, (iii) 414, (iv) 1440, (v) 5400, (vi) 2040