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Which of the following numbers is divisible by both 14 and 12?
84
126
168
210
84
Check divisibility by the factors of the numbers: 12 is 3├Ч4 and 14 is 2├Ч7. Thus, the number must be divisible by 3, 4, and 7. 84 is the smallest, hence test it first.
Divisors are 14 and 12. Options are 84, 126, 168, 210.
LCM(a,b)=HCF(a,b)a├ЧbтАЛ
Check divisibility by the factors of the numbers: 12 is 3├Ч4 and 14 is 2├Ч7. Thus, the number must be divisible by 3, 4, and 7. 84 is the smallest, hence test it first.
Many students check only one number (e.g., 14) and stop, forgetting that the number must be a common multiple of both 14 and 12.
Find Prime Factorization
Break down 14 and 12 into their prime factors to find the Least Common Multiple (LCM).
14=2├Ч7,12=22├Ч3
Calculate LCM
The LCM is the product of the highest powers of all prime factors present.
LCM(14,12)=22├Ч3├Ч7=4├Ч3├Ч7=84
Verification
A number is divisible by both 14 and 12 if and only if it is a multiple of their LCM, 84.
84├╖14=6,84├╖12=7
A is correct because 84 is the smallest number (the LCM) that is exactly divisible by both 14 and 12.
This concept is foundational for 'Time and Work' problems where you calculate the LCM of individual work days to determine total units of work.
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