bibha wrote:
Hey,
I am a new member of this club.
Could anyone please help me with few problems of quant??? Here are the problems.
1.If x is a positive integer, is the remainder 0 when 3^(x) + 1 is divided by 10?
(1) x = 4n + 2, where n is a positive integer.
(2) x > 4
2.What is the sum of a certain pair of consecutive odd integers?
(1) At least one of the integers is negative.
(2) At least one of the integers is positive.
Thanks,
Bibha
(1) x = 4n + 2, where n is a positive integer.
Last digit of
3^x repeats in blocks of 4: {3, 9, 7, 1} - {3, 9, 7, 1} - ... So cyclicity of the last digit of 3 in power is 4. Now,
3^{4n+2} will have the same last digit as
3^2 (remainder upon division 4n+2 upon cyclicity 4 is 2, which means that 3^{4n+2} will have the same last digit as 3^2). Last digit of
3^2 is
9. So
3^{4n+2}+1 will have the last digit
9+1=0. Number ending with 0 is divisible by 10 (remainder 0). Sufficient.
(2) x > 4. Clearly insufficient.
Answer: A.
Check Number Theory chapter of Math Book for more:
math-number-theory-88376.html2. What is the sum of a certain pair of consecutive odd integers?(1) At least one of the integers is negative --> infinite pairs are possible: ... (-3,-1); (-17,-15); ... (-1, 1); ... Not sufficient.
(2) At least one of the integers is positive --> infinite pairs are possible: ... (3,5); (19,21); ... (-1, 1); ... Not sufficient.
(1)+(2) one odd integer must be positive and another negative. As they are consecutive odd integers, there is only one pair possible (-1, 1) --> -1+1=0. Sufficient.
Answer: C.
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