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Find the HCF of 84, 126 and 210.
42
14
70
21
42
Find the difference between consecutive numbers (126−84=42; 210−126=84). The HCF must be a factor of the smallest difference (42). Since 42 divides 84, 126, and 210, it is the HCF.
Three numbers are provided: 84, 126, and 210.
HCF(a,b,c)=Product of common prime factors with lowest exponents
Find the difference between consecutive numbers (126−84=42; 210−126=84). The HCF must be a factor of the smallest difference (42). Since 42 divides 84, 126, and 210, it is the HCF.
Students often stop after finding a common factor like 14 or 21 without checking if it is the 'Highest' possible common factor.
Prime factorization
Break down each number into its prime components: 84=22×3×7, 126=2×32×7, and 210=2×3×5×7.
84=22⋅3⋅7,126=2⋅32⋅7,210=2⋅3⋅5⋅7
Identify common prime factors
Select the prime factors present in all three numbers with the lowest power: 21, 31, and 71.
HCF=21×31×71
Calculate final HCF
Multiply the identified factors to obtain the result.
2×3×7=42
A is correct because the prime factorization of 84, 126, and 210 shares 2×3×7, which equals 42.
This method of finding the HCF via difference is extremely useful in 'Number System' problems where you need to find the largest number that divides x,y,z leaving the same remainder.
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