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If a and b are odd integers such that a - b > 7, what is the smallest

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Bunuel
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If a and b are odd integers such that a - b > 7, what is the smallest [#permalink] New post   18 Aug 2020, 01:34
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If a and b are odd integers such that a - b > 7, what is the smallest possible positive difference between a and an even number less than b?

A. 7
B. 8
C. 9
D. 10
E. 11
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C
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Re: If a and b are odd integers such that a - b > 7, what is the smallest [#permalink] New post   18 Aug 2020, 02:19
Ques: If a and b are odd integers such that a - b > 7, what is the smallest possible positive difference between a and an even number less than b?

sol: Since a and b are both odd numbers, (a-b) will be an even number.

min(a-b) > 7
=> min(a-b) = 8

Here we can do number plugging. Let's say a = 1 and b = 9

Nearest even number less than b = 8
Diff = 8 - 1 = 7 (A)
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Re: If a and b are odd integers such that a - b > 7, what is the smallest [#permalink] New post   18 Aug 2020, 02:21
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If a and b are odd integers such that a - b > 7, what is the smallest possible positive difference between a and an even number less than b?

A. 7
B. 8
C. 9
D. 10
E. 11
Minimum difference possible between a and b is 8.
a - b = 8

As b is odd, an even number just less than b would give smallest possible positive difference. Thus, the even number is = b - 1
a - (b - 1) = 8 + 1 = 9

Answer C.
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Re: If a and b are odd integers such that a - b > 7, what is the smallest [#permalink] New post   18 Aug 2020, 02:29
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For the smallest possible positive difference between a and an even number less than b, lets take the value of b which will minimize a-b,

If b=a-8, a-b=8

Since largest even number less than b is b-1, the required difference is a-(b-1)=a-b+1=8+1=9

Answer is (C)
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Re: If a and b are odd integers such that a - b > 7, what is the smallest [#permalink] New post   18 Aug 2020, 02:48
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IMO C

Let look at an integer sequence: x, b, b+1, ... ,a
In which:
- x,a,b are odd.
- a-b >7

We want to have the smallest (a-x), in other words we want the smallest "distance" between a and x. Basically, we want the smallest a and the biggest x
- The smallest a:
a-b > 7
=> Our set will look like this:
b, b+1, b+2, ... , b+7, ... , a
=> The smallest a will be b+8.
Check: b is odd => a=b+8 will be odd => Satisfied

- The biggest x:
x is an even number smaller than b.
=> Our set: x, ..., b-2, b-1, b
=> Biggest x will be b-1
Check: b is odd => x=b-1 will be even => Satisfied.

So, "the smallest possible positive difference between a and an even number less than b" will be:
a-x = (b+8) - (b-1) = 9.
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Re: If a and b are odd integers such that a - b > 7, what is the smallest [#permalink] New post   18 Aug 2020, 03:28
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If a and b are odd integers such that a - b > 7, what is the smallest possible positive difference between a and an even number less than b?

A. 7
B. 8
C. 9
D. 10
E. 11
Solution:

Suppose:
a..........b
9..........1
11........3
13........5
.............

Given that:
a - b > 7
11(a) - 2(an even number less than b) = 9

Answer: C
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Re: If a and b are odd integers such that a - b > 7, what is the smallest [#permalink] New post   18 Aug 2020, 03:33
Explanation:

Let a=9, b=1
a-b = 8 which is greater than 7. But 1 less then B is 0 so not possible
if a=11, b=3
then, 11-3 = 8 Which is also greater than 7.(satisfies the condition of a-b>7)
So 1 less than b is 2 (which is even)
11-2 = 9

IMO-C
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Re: If a and b are odd integers such that a - b > 7, what is the smallest [#permalink] New post   18 Aug 2020, 03:57
We need to find the least difference.

It is given that a and b are odd.
Lets take a = 9 and b = 1.
Their difference is greater than 7.
9 - 0(even no. less than b)
is 9
So B .

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Re: If a and b are odd integers such that a - b > 7, what is the smallest [#permalink] New post   18 Aug 2020, 04:35
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Let 'c' be an even number less than 'b'.
Since 'a' is odd and 'c' is even the difference a - c = odd.
So option B and D are eliminated.

Now we know that a - b > 7. a and b are odd so, a - b = even. Hence he smallest difference between a and b is 8.

Since c is an even number less than b it means that if we subtract c from a then the difference will be greater than 8 so, option A is eliminated.

Now out of the remaining options 9 and 11. 9 is the smaller number.
So, option C
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Re: If a and b are odd integers such that a - b > 7, what is the smallest [#permalink] New post   18 Aug 2020, 05:13
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Quote:
If a and b are odd integers such that a - b > 7, what is the smallest possible positive difference between a and an even number less than b?

A. 7
B. 8
C. 9
D. 10
E. 11


odd-odd=even
odd-even=odd
a-b≥8
a,b=9,1 dif=8
a,even<b=9,0 dif=9

ans (C)
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Re: If a and b are odd integers such that a - b > 7, what is the smallest [#permalink] New post   18 Aug 2020, 06:41
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I will go with C
Odd-odd = even
So smallest even possible integer for a-b will be 8, therefore, smallest possible values for a and b will be 9 and 1and 0 is less than 1 and even integer so 9-0 will be 9

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Re: If a and b are odd integers such that a - b > 7, what is the smallest [#permalink] New post   18 Aug 2020, 07:05
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Bunuel wrote:
If a and b are odd integers such that a - b > 7, what is the smallest possible positive difference between a and an even number less than b?

A. 7
B. 8
C. 9
D. 10
E. 11


Let \(a=11\) and \(b=3\) this satisfies \(a−b>7\)
Now even number less than 3 is 2 ,0,-2 etc
Smallest possible positive difference is obtained when the even number is 2
hence a−2=9

IMO Ans - C
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Re: If a and b are odd integers such that a - b > 7, what is the smallest [#permalink] New post   18 Aug 2020, 07:11
c,,, as a will be 11 and b will be 3 so x has to be 2
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Re: If a and b are odd integers such that a - b > 7, what is the smallest [#permalink] New post   18 Aug 2020, 08:52
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a and b=odd integers such that a - b > 7
what is the smallest possible positive difference between a and an even number less than b?

We can suppose that: a=17 and b1=9 -> a-b1=8 (>7)
If a=17 and b2=8 (even number less than b1) -> a-b2=9

Answer -> C
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Re: If a and b are odd integers such that a - b > 7, what is the smallest [#permalink] New post   18 Aug 2020, 10:26
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If a and b are odd integers such that a - b > 7, what is the smallest possible positive difference between a and an even number less than b?

The difference should be greater than 7
So among all the options
7 can not be the answer as difference is greater than 7
8 can not be the answer as difference will be odd
So answer is 9
C
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Re: If a and b are odd integers such that a - b > 7, what is the smallest [#permalink] New post   18 Aug 2020, 14:35
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If a and b are odd integers such that a - b > 7, what is the smallest possible positive difference between a and an even number less than b?

A. 7
B. 8
C. 9
D. 10
E. 11

Odd and even happen only for positive integers, so we can ignore negative ones for this. The smallest value a van take is 9, when b = 1, so the even number less than b is 0, and the difference will be = 9 - 0 = 9

C is the answer.
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Re: If a and b are odd integers such that a - b > 7, what is the smallest [#permalink] New post   18 Aug 2020, 14:58
Let a= 11 and b =3

hence a-b>7
also a - (even number smaller than b) ie 2 =9

this will work for any pair of (a,b) where a-b = 8 (since odd-odd =even) and greatest possible even number taken ie less than b

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Re: If a and b are odd integers such that a - b > 7, what is the smallest [#permalink]
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