The integer P is greater than 7. If the integer P leaves a remainder

We can create the following equation:

P = 9Q + 4

Thus, we see that P can be values such as:

13, 22, 31, 40, ...

Let’s review the answer choices.

A) The number that is 4 less than P is a multiple of 9.

P - 4 = 9Q + 4 - 4

P - 4 = 9Q

A is true.

B) The number that is 5 more than P is a multiple of 9.

P + 5 = 9Q + 4 + 5

P + 5 = 9Q + 9

B is true.

C) The number that is 2 more than P is a multiple of 3.

P + 2 = 9Q + 4 + 2

P + 2 = 9Q + 6

C is true.

D) When divided by 3, P will leave a remainder of 1.

(9Q + 4)/3 = (3Q + 1) + 1/3

D is true.

E) When divided by 2, P will leave remainder of 1.

(9Q + 2)/2 = (4Q + 1) + Q/2

We can’t be certain what the remainder is. If Q = 1, then the remainder is 1; however, if Q is 2, then the remainder is 0.

Alternate solution:

Since the problem says all of the answer choices must be true except one of them, we can use any integer > 7 that satisfies the condition “when it is divided by 9, it will leave a remainder of 4” to check the answer choices. Since 13 is one such number, we will use that.

A) 13 - 4 = 9 and 9 is a multiple of 9. This is true.

B) 13 + 5 = 18 and 18 is a multiple of 9. This is true.

C) 13 + 2 = 15 and 15 is a multiple of 3. This is true.

D) 13/3 = 4 R 1. This is true.

E) 13/2 = 6 R 1. This is true.

Since 13 makes all answer choices true, we need to use another number. Another number we can use is 13 + 9 = 22.

Instead of checking each answer choice again, we can see that choice E is not true, since when 22 is divided by 2, the remainder is 0, not 1.

Answer: E

E is the correct option .

P = 9k + 4

k= 1,2,3 ....

P1 = 9*1 + 4 = 13, when divided by 2, remainder is 1

P2 = 9*2 + 4 = 22, when divided by 2, remainder is 0

if P+4 is even, e.g., 22,40,58...
then P/2 will leave no remainder
E

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