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# If 7(a – 1) = 17(b – 1), and a and b are both positive

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Retired Moderator
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If 7(a – 1) = 17(b – 1), and a and b are both positive [#permalink]  23 May 2017, 01:20
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Expert's post
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Question Stats:

50% (03:30) correct 50% (01:25) wrong based on 10 sessions
If 7(a – 1) = 17(b – 1), and a and b are both positive integers the product of which is greater
than 1, then what is the least possible sum of a and b?
A. 2
B. 7
C. 17
D. 24
E. 26

Drill 1
Question: 11
Page: 338-339

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Sandy
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Retired Moderator
Joined: 07 Jun 2014
Posts: 4803
GRE 1: Q167 V156
WE: Business Development (Energy and Utilities)
Followers: 171

Kudos [?]: 2926 [3] , given: 394

Re: If 7(a – 1) = 17(b – 1), and a and b are both positive [#permalink]  04 Jun 2017, 15:01
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Expert's post
Explanation

Because 7 and 17 are prime and have no common factor greater than 1, their least common multiple will be their product: 7 × 17 = 119.

The least possible value for (a – 1), then, is 17, so a = 18; likewise, the least possible value for (b – 1) is 7, so b = 8.

The least possible sum for a and b, therefore, is 18 + 8 = 26.

Be careful if you selected choice A: Although Plugging In a value of 1 for both a and b would yield 0 on both sides of the equation, the
problem specifies that the product of a and b be greater than 1.

Hence option E is correct!
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Sandy
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Re: If 7(a – 1) = 17(b – 1), and a and b are both positive [#permalink]  29 May 2018, 09:08
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sandy wrote:
Explanation

Because 7 and 17 are prime and have no common factor greater than 1, their least common multiple will be their product: 7 × 17 = 119.

The least possible value for (a – 1), then, is 17, so a = 18; likewise, the least possible value for (b – 1) is 7, so b = 8.

The least possible sum for a and b, therefore, is 18 + 8 = 26.

Be careful if you selected choice A: Although Plugging In a value of 1 for both a and b would yield 0 on both sides of the equation, the
problem specifies that the product of a and b be greater than 1.

Hence option E is correct!

Can you please explain me the highlighted line.
Retired Moderator
Joined: 07 Jun 2014
Posts: 4803
GRE 1: Q167 V156
WE: Business Development (Energy and Utilities)
Followers: 171

Kudos [?]: 2926 [1] , given: 394

Re: If 7(a – 1) = 17(b – 1), and a and b are both positive [#permalink]  29 May 2018, 10:33
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KUDOS
Expert's post
We have $$7(a - 1) = 17(b - 1)$$.

This can be written as:

$$7(a - 1) = 17(b - 1)$$
$$7 \times 17 = 17 \times 7$$. This is the minimum possible value.

So $$(a - 1)=17$$ and $$(b - 1)=7$$

It could also be:

$$7(a - 1) = 17(b - 1)$$
$$7 \times 34 = 17 \times 14$$ which is $$7 \times 17 \times 2 = 17 \times 7 \times 2$$.

This is $$(a - 1)=34$$ and $$(b - 1)=14$$ also possible but not the minimum possible value.
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Re: If 7(a – 1) = 17(b – 1), and a and b are both positive   [#permalink] 29 May 2018, 10:33
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