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If a and b are positive integers and ab> 13, IS a≠b?

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If a and b are positive integers and ab> 13, IS a≠b?  [#permalink]

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New post 25 Jul 2018, 19:52
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If a and b are positive integers and ab> 13, IS a≠b?

1. Exactly three of the numbers ab-4, ab-2,ab+1 & ab+3 have the same ten's digit
2. ab is divisible by 8, but not divisible by 16.

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Re: If a and b are positive integers and ab> 13, IS a≠b?  [#permalink]

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New post 25 Jul 2018, 20:07
If a and b are positive integers and ab> 13, IS a≠b?

1. Exactly three of the numbers ab-4, ab-2,ab+1 & ab+3 have the same ten's digit
Say the numbers are ab-4,ab-2,ab+1 and ab+3 is not..
ab+3 can have units digit as 0 or 1 since if it is 2, even ab+1 will have same tens digit.
So if ab+3 has units digit as 0 or 1, units digit of ab will be 3 less that is 7 or 8 but no perfect square can have its units digit as 7 or 8 so a≠b as a=b means ab is a perfect square.
Now if ab-2, ab+1 and ab+3 have same tens digit, then ab-4 can have units digit as 8 or 9.
Here units digit of ab will be 4 more that is 8+4 or 9+4 hence 2 or 3, but again no perfect square has its units digit as 2 or 3..
Thus in both cases a≠b
Sufficient

2. ab is divisible by 8, but not divisible by 16.
If a=b, ab will be a perfect square hence the power of prime number will be even but here power of 2 is 3..8=2^3
Hence a≠b
Sufficient

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If a and b are positive integers and ab> 13, IS a≠b?  [#permalink]

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New post 18 Aug 2018, 01:43
ab-4,ab-2,ab+1,ab+3
ab-4=23,33,43,... > ab=27,37,47,...
or
ab+3=26,36,46,... > ab=23,33,43,...

2)
multiples of 8:
8*1,8*2,8*3,8*4,8*5,8*6,8*7,...
every second of these will be multiples of 16 since the mulitiplier is even (2,4,6,8...)
for the rest, the multiplier (1,3,5..) is odd and does not contain 2 as a factor. hence 8 cannot be root square'd

as we know, the last digit of a square is always 1,4,9,6,5. hence ab cannot be a square or a is not equal to b.
hope that helps
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If a and b are positive integers and ab> 13, IS a≠b?   [#permalink] 18 Aug 2018, 01:43
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If a and b are positive integers and ab> 13, IS a≠b?

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