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# If 8^a+8^b=3^(b-2)*8^(a-1), a and b are positive integers, then what

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If 8^a+8^b=3^(b-2)*8^(a-1), a and b are positive integers, then what  [#permalink]

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25 Nov 2019, 01:29
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If $$8^a+8^b=3^{b-2}*8^{a-1}$$, a and b are positive integers, then what is the product of a and b ?

A. 10
B. 12
C. 15
D. 18
E. 20

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Joined: 19 Oct 2018
Posts: 1297
Location: India
If 8^a+8^b=3^(b-2)*8^(a-1), a and b are positive integers, then what  [#permalink]

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Updated on: 25 Nov 2019, 09:45
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$$8^a+8^b=3^{b-2}*8^{a-1}$$

$$8^b(1+8^{a-b})=3^{b-2}*8^{a-1}$$

b=a-1 or a-b=1

3^{b-2}=1+8^1
b=4

and a= 1+4=5

a*b=20

Bunuel wrote:
If $$8^a+8^b=3^{b-2}*8^{a-1}$$, a and b are positive integers, then what is the product of a and b ?

A. 10
B. 12
C. 15
D. 18
E. 20

Are You Up For the Challenge: 700 Level Questions

Originally posted by nick1816 on 25 Nov 2019, 03:52.
Last edited by nick1816 on 25 Nov 2019, 09:45, edited 1 time in total.
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If 8^a+8^b=3^(b-2)*8^(a-1), a and b are positive integers, then what  [#permalink]

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25 Nov 2019, 09:27
1
Bunuel wrote:
If $$8^a+8^b=3^{b-2}*8^{a-1}$$, a and b are positive integers, then what is the product of a and b ?

A. 10
B. 12
C. 15
D. 18
E. 20

$$(a,b)=pos.integers$$

$$8^a+8^b=3^{b-2}*8^{a-1}…8^a+8^b=3^{b-2}*8^a/8…8(8^a+8^b)=3^{b-2}*8^a$$
$$8(8^a+8^b)/8^a=3^{b-2}…8^{a+1-a}+8^{b+1-a}=3^{b-2}…8+8^{b-a+1}=3^{b-2}$$

$$8+?=odd…even+odd=odd…8^{b-a+1}=odd=1…8+1=9$$

$$3^{b-2}=9=3^2…b-2=2…b=4$$
$$8^{b-a+1}=1…b-a+1=0…[4]-a+1=0…a=5…ab=5*4=20$$

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If 8^a+8^b=3^(b-2)*8^(a-1), a and b are positive integers, then what  [#permalink]

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25 Nov 2019, 10:00
$$8^a+8^b=3^{b-2}*8^{a-1}$$
$$8^{a-1}*(8+8^{b-a+1})=3^{b-2}*8^{a-1}$$

Since a is positive integer,
$$8+8^{b-a+1}=3^{b-2}$$

$$8+8^{b-a+1}$$ has to be a multiple of 3 (see right hand side equation). So, the only possible solution is that 8+1=9.

Therefore,
$$9=3^{b-2} -> b=4,$$ and
$$8+8^{4-a+1}=9 -> a=5,$$ and finally,
$$a.b=20$$

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If 8^a+8^b=3^(b-2)*8^(a-1), a and b are positive integers, then what   [#permalink] 25 Nov 2019, 10:00
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