Prep-T1-35(DS)

[邁向高等學府]

35.
If p and n are positive integers and p>n, What is the remainder when p^2-n^2 is divided by 15?

(1)The remainder when p+n is divided by 5 is 1
(2) The remainder when p-n is divided by 3 is 1


ANS. E

How to prove the both data is not suffieicent??

感謝大大…

[eced]

p^2-n^2=(p+n)(p-n) 推敲可能的情況 ; 15=3*5
(1)p+n 除5餘1 , 知p+n不是5的倍數,
p-n是15的倍數? 不知
p+n是3的倍數, p-n是5的倍數? 不知
(2)p-n除3餘1,知p-n不是3的倍數
p+n是15的倍數? 不知
p-n是5的倍數, p-n是3的倍數? 不知

(1)+(2)
p+n 除5餘1 , 知p+n不是5的倍數,
p-n除3餘1,知p-n不是3的倍數
那p-n是5的倍數,p+n是3的倍數? 也不知

or
p+n=5a+1; p-n=3b+1
(p+n)(p-n) =15ab+[ 5a+3b+1 ]
5a+3b+1 可能是15的倍數 也可能不是