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Answer :
To find the Highest Common Factor (H.C.F.), also known as the Greatest Common Divisor (G.C.D.), of the given numbers, we can use the Euclidean algorithm, which is an efficient method based on division.
Let's go through each part step-by-step:
(b) 198 and 360
Divide 360 by 198:
[tex]360 \div 198 = 1 \text{ quotient with a remainder of } 162[/tex]
Divide 198 by 162:
[tex]198 \div 162 = 1 \text{ quotient with a remainder of } 36[/tex]
Divide 162 by 36:
[tex]162 \div 36 = 4 \text{ quotient with no remainder}[/tex]
Since the remainder is 0, the divisor at this step, 36, is the H.C.F. of 198 and 360.
(e) 2684 and 1098
Divide 2684 by 1098:
[tex]2684 \div 1098 = 2 \text{ quotient with a remainder of } 488[/tex]
Divide 1098 by 488:
[tex]1098 \div 488 = 2 \text{ quotient with a remainder of } 122[/tex]
Divide 488 by 122:
[tex]488 \div 122 = 4 \text{ quotient with no remainder}[/tex]
Since the remainder is 0, the divisor at this step, 122, is the H.C.F. of 2684 and 1098.
(h) 1456, 1183, and 3640
To find the H.C.F. of three numbers, we first find the H.C.F. of two of them, then use that result with the third number.
Find the H.C.F. of 1456 and 1183:
Divide 1456 by 1183:
[tex]1456 \div 1183 = 1 \text{ quotient with a remainder of } 273[/tex]
Divide 1183 by 273:
[tex]1183 \div 273 = 4 \text{ quotient with a remainder of } 91[/tex]
Divide 273 by 91:
[tex]273 \div 91 = 3 \text{ quotient with no remainder}[/tex]
Since the remainder is 0, the divisor at this step, 91, is the H.C.F. of 1456 and 1183.
Find the H.C.F. of 91 and 3640:
Divide 3640 by 91:
[tex]3640 \div 91 = 40 \text{ quotient with a remainder of } 0[/tex]
Since the remainder is 0, the divisor, 91, is also the H.C.F. of the group 1456, 1183, and 3640.
By following these steps, we find that the H.C.F. of 198 and 360 is 36, the H.C.F. of 2684 and 1098 is 122, and the H.C.F. of 1456, 1183, and 3640 is 91.
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