The digits of a three digit number A are written in the reverse order to form another three digit number B. If B > A and B – A is perfectly divisible by 7, then which of the following is necessarily true?
1. 100 < A < 299 2. 106 < A < 305
3. 112 < A < 311 4. 118 < A < 317
correct answer is 2. 106 < A < 305
If the digits of A are abc, Then B is cba.
100c+10b+a - 100a+10b+c = 7k
99c-99a = 7k
99(c-a) = 7k
so, c-a is a multiple of 7. In fact, since c and a are both single digits, c-a=7.
so, A must begin with 1 or 2, making c either 8 or 9
So, of the given choices, 100<A<299
Hmmm.
Since A=101,102,103,104,105,300,301,302,303,304 all don't work,
Both answers are true.
Since actually 107 works and 299 works,
107 <= A <= 299
You are right. I should not have excluded 299 from the solution set.
To solve this problem, let's break it down step by step:
Step 1: Determine the possible values of A.
Since A is a three-digit number, the first digit cannot be zero. The largest possible three-digit number formed in reverse order is 999, so the largest possible value for A is 999. Therefore, A can only take values from 100 to 999.
Step 2: Determine the possible values of B.
B is formed by reversing the digits of A. For example, if A = 123, then B = 321. Since B > A, we can conclude that the hundreds digit of B must be greater than the hundreds digit of A. If the hundreds digit of A is x, then the hundreds digit of B must be greater than x. Similarly, the tens digit of B must be greater than the tens digit of A, and the units digit of B must be greater than the units digit of A.
Step 3: Determine the condition for B – A to be divisible by 7.
The difference between B and A can be found by subtracting A from B: B – A. If B – A is divisible by 7, it means that the remainder when (B – A) is divided by 7 is zero. In other words, (B – A) % 7 = 0, where % denotes the modulo operation.
Step 4: Analyze the answer choices.
Let's check each of the answer choices to see if they are necessarily true:
1. 100 < A < 299: A can be any value from 100 to 999, so this statement is not necessarily true.
2. 106 < A < 305: A can be any value from 100 to 999, so this statement is not necessarily true.
3. 112 < A < 311: A can be any value from 100 to 999, so this statement is not necessarily true.
4. 118 < A < 317: A can be any value from 100 to 999, so this statement is not necessarily true.
Based on our analysis, none of the answer choices are necessarily true.