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Sunday, May 3, 2009
Problem 18 - GMAT Number Properties
If positive integer x is divided by 5, the result is p and the remainder 3. If x is divided by 11, the remainder is 3 again, what is the remainder when p is divided by 11?
For those of you, who are not aware of modular arithmetic - let me explain what Blaoism did. a=b(mod n) by definition (a-b)/n must be an integer. (Modulo definition) In division problems we can use the following congruence notation: 38=2(mod6) all this means is 38-2/6 is an integer. In reality we must use the (equivalence or the congruence sign with 3-horizontal parallel bars as opposed to the 2 in the equal to"=" sign).
For this problem: x=5p+3 = 3(mod11) =3(mod5) 5p+3=3(mod11) or (5p+3-3)/11 must be an Integer or 5p/11 must be an integer .This implies p must be divisible by 11 or p/11 is an integer. or the form: 5p=0(mod11) or p=0(mod)11 Hence remainder 0
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x = 5p +3 =3 (mod 11)
ReplyDelete5p = 0 mod (11)
so, p = 0 (mod 11), since (5,11) = 1
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ReplyDeleteFor those of you, who are not aware of modular arithmetic - let me explain what Blaoism did.
ReplyDeletea=b(mod n) by definition (a-b)/n must be an integer. (Modulo definition)
In division problems we can use the following congruence notation:
38=2(mod6) all this means is 38-2/6 is an integer.
In reality we must use the (equivalence or the congruence sign with 3-horizontal parallel bars as opposed to the 2 in the equal to"=" sign).
For this problem:
x=5p+3 = 3(mod11) =3(mod5)
5p+3=3(mod11) or
(5p+3-3)/11 must be an Integer
or 5p/11 must be an integer .This implies p must be divisible by 11 or p/11 is an integer.
or the form:
5p=0(mod11) or p=0(mod)11
Hence remainder 0