【文章內(nèi)容簡介】 than 3, square it and see what would be the remainder upon division of it by 12.n=5 n^2=25 remainder upon division 25 by 12 is 1.Answer:B.Pattern3:min/max question involving remaindersWhen positive integer n is divided by 5, the remainder is 1. When n is divided by 7, the remainder is 3. What is the smallest positive integer k such that k + n is a multiple of 35.A. 3B. 4C. 12D. 32E. 35Sol：Positive integer n is divided by 5, the remainder is 1 n = 5q+1, where q is the quotient 1, 6, 11, 16,21, 26,31, ...Positive integer n is divided by 7, the remainder is 3 n=7p+3, where p is the quotient 3, 10, 17, 24, 31,....You cannot use the same variable for quotients in both formulas, because quotient may not be the same upon division n by two different numbers.For example 31/5, quotient q=6 but 31/7, quotient p=4.There is a way to derive general formula for n (of a type n= mx+r, where x is divisor and r is a remainder)based on above two statements:Divisor x would be the least mon multiple of above two divisors 5 and 7, hence x=35.Remainder r would be the first mon integer in above two patterns, hence r=31.Therefore general formula based on both statements is n=35m+ the smallest positive integer k such that k+n is a multiple of 35 is 4 n+4 = 35k+31+4 = 35(k+1).Answer:BPattern4:disguised PS remainder problem.There are between 100 and 110 cards in a collection of cards. If they are counted out 3 at a time, there are 2 left over, but if they are counted out 4 at a time, there is 1 left over. How many cards are in the collection?(A) 101(B) 103(C) 106(D) 107(E) 109Sol：If the cards are counted out 3 at a time, there are 2 leftover: x=3q+2. From the numbers from 100 to 110 following three give the remainder of 2 upon division by 3: 101, 104 and 107。If the cards are counted out 4 at a time, there are 1 leftover: x=4p+1. From the numbers from 100 to 110 following three give the remainder of 1 upon division by 4: 101, 105 and 109.Since x, the number of cards, should satisfy both conditions then it equals to 101.Answer:APattern5:we need to answer some question about an integer, when the statements give info involving remainders.Q1：OG13 Practice Questions, question 58What is the tens digit of positive integer x ?(1) x divided by 100 has a remainder of 30.(2) x divided by 110 has a remainder of 30.Sol：(1) x divided by 100 has a remainder of 30 x =100q+30, so x can be: 30, 130, 230, ... Each has the tens digit of 3. Sufficient.(2) x divided by 110 has a remainder of 30 x =110p+30, so x can be: 30, 250, ... We already have two values for the tens digit. Not sufficient.Answer:A.Q2：OG13 Practice Questions, question 83If k is an integer such that 56 k 66, what is the value of k ?(1) If k were divided by 2, the remainder would be 1.(2) If k + 1 were divided by 3, the remainder would be 0.Sol：(1) If k were divided by 2, the remainder would be 1 k is an odd number, thus it could be 57, 59, 61, 63, or 65. Not sufficient.(2) If k + 1 were divided by 3, the remainder would be 0 k is 1 less than a multiple of 3, thus it could be 59, 62, or 65. Not sufficient.(1)+(2) k could still take more than one value: 59 or 65. Not sufficient.Answer:E.Pattern6:we need to find the remainder when some variable or an expression with variable(s) is divided by some integer. Usually the statements give divisibility/remainder info.Most mon patter.Q1：What is the remainder when the positive integer n is divided by 6?(1) n is multiple of 5(2) n is a multiple of 12Sol：(1) n is multiple of 5. If n=5, then n yields the remainder of 5 when divided by 6 but if n=10, then n yields the remainder of 4 when divided by 6. We already have two different answers, which means that this statement is not sufficient.(2) n is a multiple of 12. Every multiple of 12 is also a multiple of 6, thus n divided by 6 yields the remainder of 0. Sufficient.Answer:B.Q2：If x and y are integer, what is the remainder when x^2 + y^2 is divided by 5?(1) When xy is divided by 5, the remainder is 1(2) When x+y is divided by 5, the remainder is 2Sol：(1) When xy is divided by 5, the remainder is 1 xy = 5q+1, so xy can be 1, 6, 11, ... Now, x=2 and y=1 (xy=1) then x^2+y^2= 5 and thus the remainder is 0, but if x=3 and y=2 (xy=1)then x^2+y^2= 13 and thus the remainder is 3. Not sufficient.(2) When x+y is divided by 5, the remainder is 2 x+y=5p+2, so x+y can be 2, 7, 12, ... Now, x=1 and y=1 (x+y=2) then x^2+y^2= 2 and thus the remainder is 2, but if x=5 and y=2 (x+y=7) then x^2+y^2= 29 and thus the remainder is 4. Not sufficient.(1)+(2) Square both expressions: x^2 2xy +y^2= 25q^2 + 10q+ 1 and x^2 +2xy +y^2= 25p^2 + 20p + 4 add them up: 2(x^2+y^2)= 5(5q^2+2q+5p^2+4p+1) so 2(x^2+y^2) is divisible by 5 (remainder 0), which means that so is x^2+y^2. Sufficient.Answer:C.Q3：If t is a positive integer and r is the remainder when t^2+5t+6 is divided by 7, what is the value of r?