Find the greatest number , which will divide 215, 167 and 135 so as to leave the same remainder in each case
A. 64
B. 32
C. 24
D. 16
Notice that
the remainder we get when we divide N by d is the same as when we divide N+d by d.
For example, when we divide 16 by 7, we get remainder 2. When we divide (16+7) by 7, we get remainder 2.
Likewise, when we divide 53 by 10, we get remainder 3. When we divide (53+10) by 10, we get remainder 3.
We can extend this property to say that
the remainder we get when we divide N by d is the same as when we divide N+kd by d (where k is some integer) In other words, N and N+kd will have the same remainder when we divide both by d
We can take all of this information and conclude that,
If A > B, and A and B have the same remainder when divided by d, then A - B = some multiple of d. In other words, d is a
divisor of (A-B)
Okay, now onto the question. . .
Our three numbers are 135, 167, and 215
Notice that 167 is
32 greater than 135 (167 - 135 =
32).
So, in order for 135 and 167 to have the same remainder, we must divide both by a number that is a divisor of
32
Since 64 and 24 are NOT divisors of
32, we can eliminate A and C
Also notice that 215 is
48 greater than 167 (215 - 167 =
48)
So, in order for 167 and 215 to have the same remainder, we must divide both by a number that is a divisor of
48
Since 32 is NOT a divisor of
48, we can eliminate B
This leaves us with
D, the correct answer.
Cheers,
Brent
