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If the LCM of two numbers 36 and 48 is expressed as 2с╡Г * 3с╡З, find the value of a + b.
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Express both numbers as product of primes: 36=22imes32 and 48=24imes31. Take the highest power of each prime present: 24 and 32.
Two numbers are 36 and 48. Their LCM is in the form 2с╡Г * 3с╡З.
LCM(a,b)=тИПpimax(eiтАЛ)тАЛ
Express both numbers as product of primes: 36=22imes32 and 48=24imes31. Take the highest power of each prime present: 24 and 32.
Students often pick the sum of the powers of the individual numbers (2+2=4 and 4+1=5) instead of finding the LCM first by taking the maximum exponent of each prime factor.
Prime Factorization of 36
Divide 36 by prime numbers: 36=6imes6=(2imes3)imes(2imes3)=22imes32.
36=22imes32
Prime Factorization of 48
Divide 48 by prime numbers: 48=16imes3=24imes31.
48=24imes31
Calculate LCM
Take the highest exponent of each prime factor: a=max(2,4)=4 and b=max(2,1)=2.
LCM=24imes32
Calculate a + b
Using the exponents found, calculate the sum: a+b=4+2=6.
a+b=4+2=6
B is correct because the prime factorization of LCM(36,48) is 24imes32, making a=4 and b=2, so a+b=6.
This concept of prime power representation is essential for finding the number of divisors of a number, calculated as (e1тАЛ+1)(e2тАЛ+1)...(enтАЛ+1).
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