Problem Solving

If s and t are positive integers such that s/t=64.12,which of the following could be the remainder when s is divided by t?
A) 2
B) 4
C) 8
D) 20
E) 45

Let's examine a few things about remainders and decimal conversions.
7/4 = 1 3/4 = 1.75. When we divide 7 by 4, the remainder is 3, and .75 = 3/4.
32/5 = 6 2/5 = 6.4. When we divide 32 by 5, the remainder is 2, and .4 = 2/5.
58/20 = 2 18/20 = 2.9. When we divide 58 by 20, the remainder is 18, and .9 = 18/20.
As you can see, there is an important relationship between the remainder and the decimal part of the conversion.

64.12 = 64 12/100 = 6412/100. So, it's possible that s/t = 6412/100, in which case the remainder is 12 when s is divided by t.
Check the answer choices. . . nope, 12 is not one of the options.

Also, recognize that 64.12 = 64 12/100 = 64 3/25 = 1603/25. So, it's possible that s/t = 1603/25, in which case the remainder is 3 when s is divided by t.
Check the answer choices. . . nope, 3 is not one of the options.

At this point, we should recognize that we can get ANY MULTIPLE OF 3 as the remainder.
For example, 64.12 = 64 12/100
= 64 3/25
= 64 6/50
= 3206/50 = s/t, in which case the remainder is 6 when s is divided by t.

Or...64.12 = 64 12/100
= 64 3/25
= 64 9/75
= 4809/75 = s/t, in which case the remainder is 9 when s is divided by t.

And so on.

Since only one answer choice (E) is A MULTIPLE OF 3, E must be the correct answer.

Aside: Here's further proof:
64.12 = 64 12/100
= 64 3/25
= 64 45/375
= 24045/375 = s/t, in which case the remainder is 45 when s is divided by t.

Cheers,
Brent

yasirnasir wrote:Hi, I cannot understand this remainder question solution from official guide. Can anybody help me? I just restarted my prep and I am completely rusty about most of the concepts.

13. If s and t are positive integers such that ,
which of the following could be the remainder when
s is divided by t ?
(A) 2
(B) 4
(C) 8
(D) 20
(E) 45 $$$$ $$$$

This problem will be best solved using the remainder formula. Let's first state the remainder formula:

When positive integer x is divided by positive integer y, if integer Q is the quotient and r is the remainder, then x/y = Q + r/y.

In this problem, we are given the following:

s/t = 64.12

We can simplify this to read as the remainder formula:

s/t = 64 + 0.12

s/t = 64 + 0.12

s/t = 64 + 12/100

s/t = 64 + 3/25

Because Q is always an integer, we see that Q must be 64, and thus the remainder r/y must be 3/25. We can now equate r/y to 3/25 and determine a possible value for r.

r/y = 3/25

Note that some equivalent values for r/y could be 6/50 or 9/75 or 12/100, and so forth. Note that in all cases, the value of r is a multiple of 3.

Of the answer choices, the only multiple of 3 is 45, so that is a possible value of r.

Answer: E