Answer the following multiple choice questions?

2015-08-28 5:14 pm
If j, k, and n are consecutive integers such that 0< j<k<n and the units (ones) digit of the product jn is 9, what is the units digit of k ?
(A) 0 (B) 1 (C) 2 (D) 3 (E) 4
i have the answer i need the logic explained thanks guys

回答 (2)

2015-08-28 5:43 pm
✔ 最佳答案
Because they are consecutive and k is the
one in the middle:

j = (k - 1) and n = (k + 1)

The product jn in terms of k is then

(k-1)(k+1) = k^2 - 1

Since k is positive we can say that k can be
expressed as 10d + u, where u is the units digit
of k

and so we have

(10d + u)^2 - 1

100d^2 + 20ud + u^2 - 1

100d^2 + 20ud is a multiple of 10

and so we have

Some multiple of 10 + u^2 - 1

and we can see what sort of units digit will
result for diffent units digits.

If (u^2 - 1) is positive the units digit will be
the units digit from u^2 - 1

And so we can see that 1 will get you a units
digit of 0, 2 will result in a units digit of 3,
3 will get you 8, and 4 will get you 5

But if the units digit is 0, then

(u^2 - 1) is -1

and so you will have

"some multiple of 10" - 1, will always has
a units digit of 9.

The units digit of k must be 0

So for example if k = 10
then j = 9 and n = 11 and jn = 99
2015-08-28 5:25 pm
j, k and n are consecutive integers.
j = n ‒ 2

Units digit of n : 0 1 2 3 4
Units digit of j : 8 9 0 1 2

1 × 9 = 9
The units digit of j = 9
The units digit of n = 1

j, k and n are consecutive integers.
The units digit of k = 0 ...... The answer is (A).


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