Do not multiply this out at home

Number Theory Level 1

The least common multiple of $2^ {12} \times 3^{32} \times 5^{54}$ and $2^{21} \times 3^{23} \times 5^{45}$ can be written as $2^a \times 3^b \times 5^c$ , where $a, b$ and $c$ are non-negative integers. What is the value of $a+b+c$ ?

107 80 89 98

7 solutions

Ajay Maity
Dec 1, 2013

LCM of a^b and a^c (considering b>c) is a^b. in this case LCM of 2^12 and 2^21 is 2^21, so a is 21. LCM of 3^32 and 3^23 is 3^32, so b is 32. LCM of 5^54 and 5^45 is 5^45, so c is 54. Hence, a + b + c = 21 + 32 + 54 = 107.

ahh missed it lcm(12,21)=84 <--- etc... trick question?

Dewayne Gunter - 7 years, 6 months ago

Dewayne, you are right. LCM of 12 and 21 is 84. But here, we need to find the LCM of 2^12 and 2^21.

I am sure you found the LCM of 12 and 21 as follows:

12 = 3 x 2 x 2 = 3^1 x 2^2

21 = 3 x 7 = 3^1 x 7^1

And then you took the highest powers of all the prime numbers:

i.e. LCM = 3^1 x 2^2 x 7^1 = 3 x 4 x 7 = 84

Same goes here:

LCM of 2^12 and 2^21 would be 2^12.

Remember 12 and 21 is in power! Hope you understood.

Ajay Maity - 7 years, 6 months ago

oh... i have wrong addition of the greater exponents....

Darius Cana - 7 years, 6 months ago

Karthik Dayal
Dec 20, 2013

Take the max. powers of every term and add. we get 21+32+54 =107

i think this method is wrong he asked the LCM so substarct the powers =(21-12)+(32-23)+(54-45) =9+9+9+ =27

chiranjeevi rao Chitrothu - 7 years, 5 months ago

Andhika Rahardian
Jan 12, 2014

Just Find The largest powers of each bases like

$2^{12}$ with $2^{21}$ = $21$

$3^{23}$ with $3^{32}$ = $32$

$5^{45}$ with $5^{54}$ = $54$

$21+32+54$ = $\boxed{107}$

multiply this on home will get your calculator screaming

Rajasekhar E
Feb 22, 2014

The LCM of any number = Product of common factors X Remaining factors so that here LCM = 2^21x3^32x5^54. Therefore a=21,b=32,c=54 So the value of a+b+c=21+32+54=107

Dilbwag Singh
Feb 7, 2014

The LCM of the numbers will be equal to 2^21x3^32x5^54. Therefore a=21,b=32,c=54 So the value of a+b+c=21+32+54=107

Shaik Himavali
Dec 4, 2013

take highest power valued number , then, will get l.c.m as 2 power of 21 * 3 power of 32 * 5 power of 54 . therefore a =21 ,b=32 and c=54. a+b+c=107

B.S.Bharath Sai Guhan
Dec 4, 2013

L.C.M is defined as the product of highest powers of primes in the numbers' prime factorization. In this case, that is $2^{21}\times 3^{32} \times 5^{54}$ . Hence we get a =21, b =32 and c =54

Do not multiply this out at home

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