x and y are positive integers. If the given expression

Question

x and y are positive integers. If the given expression 864 × \(14^{14x}\) + 93 × 47\(^{12y}\) when divided by 5 leaves remainder p and when divided by 9 leaves remainder q, then what is the remainder when p + q is divided by 11?

A. 2

B. 3

C. 5

D. 10

E. Cannot be determined

A. 2

B. 3

C. 5

D. 10

E. Cannot be determined

gmatophobia · Answer

Step 1  :  Calculate the value of p

864 × 14^{14x} + 93 × 47^{12y}

We can find the remainder, p, by finding the value of the unit digit of the number

UD → Units Digit of

UD( 14^{14x} ) = 6 (used the concept of cyclicity of 4)

UD( 47^{12y} ) = 1 (used the concept of cyclicity of 7)

UD ( 864 × 14^{14x} + 93 × 47^{12y} ) = UD( 4*6 + 3*1 ) =  7

Remainder( \frac{7}{5} ) =  p = 2

Step 2  :  Calculate the value of q

864 × 14^{14x} + 93 × 47^{12y}

864 is divisible by 9, hence the remainder of the term  864 × 14^{14x}  = 0

Remainder ( \frac{93}{9} ) = 3

Remainder ( \frac{47}{9} ) = 2

Remainder( \frac{3 × 2^{12y}}{9} )

Remainder( \frac{3 × 8^{4y}}{9} )

Remainder( \frac{3 }{9} )* Remainder( \frac{8^{4y} }{9} ) =  3 * 1  = 3

q = 3

Step 3  :  Remaining Calculations

p + q = 2 + 3 = 5

Remainder( \frac{p + q}{11} ) = Remainder( \frac{5}{11} ) = 5

Option C

geovanni · Answer

864x1414 + 93x4712y

We can rewrite this expression as:

(864x1414) + (93x4712y)

Now, we need to find the remainders of each term when divided by 5 and 9.

Remainder when divided by 5:

864x1414: The last digit of 864 is 4, and the last digit of 1414 is 4. So, the last digit of their product is 4 * 4 = 16, which has a last digit of 6. Therefore, the remainder when 864x1414 is divided by 5 is 6.
93x4712y: The last digit of 93 is 3, and the last digit of 4712y is y. So, the last digit of their product is 3 * y. Since y is a positive integer, the last digit of 3 * y can be any digit from 0 to 9. Therefore, the remainder when 93x4712y is divided by 5 can be any digit from 0 to 9.
Remainder when divided by 9:

864x1414: The sum of the digits of 864 is 8 + 6 + 4 = 18, which is divisible by 9. So, the remainder when 864x1414 is divided by 9 is 0.
93x4712y: The sum of the digits of 93 is 9 + 3 = 12, which is divisible by 9. The sum of the digits of 4712y is 4 + 7 + 1 + 2 + y = 14 + y. So, for the remainder to be 0, y must be 5. Therefore, the remainder when 93x4712y is divided by 9 is 0 if y = 5.
Combining the remainders:

If y = 5, then the remainder when 864x1414 + 93x4712y is divided by 5 is 6, and the remainder when it is divided by 9 is 0.
If y ≠ 5, then the remainder when 864x1414 + 93x4712y is divided by 5 can be any digit from 0 to 9, and the remainder when it is divided by 9 is 0 only if y = 5.
Finding the remainder of p + q when divided by 11:

If y = 5, then p = 6 and q = 0. So, p + q = 6 + 0 = 6, which has a remainder of 6 when divided by 11.
If y ≠ 5, then either p is not 6 or q is not 0. In either case, p + q will not be 6, so the remainder when p + q is divided by 11 will not be 6.
Therefore, the remainder when p + q is divided by 11 is 6 if y = 5, and cannot be determined if y ≠ 5.

=> the answer is A. 2.

Chrisharley · Answer

Is there a shorter way to solve this. I understand the explanation but it takes too much time. 
Thanks.

Krunaal · Answer

864 ×  14^{14x}  + 93 × 47 ^{12y}

Let's divide by 5 and write the remainders for each value

4 x 1 + 3 x 1 (since 864%5 = 4;  14^{14x} %5 =  (-1)^{14x}  = 1; 93%5 = 3;  47^{12y} %5 =  2^{12y} %5 =  16^{3y} %5 = 1)

7%5 = 2

So p = 2

Now similarly divide by 9,

0 x whatever + 3 x 1 (since 8+6+4 = 18 is divisible by 9, remainder must be 0;  14^{14x}  => we must not be bothered with calc. this since it is multiplied by 0; 93%9 = 3;  47^{12y} %9 =  2^{12y} %9 =  8^{4y} %9 =  (-1)^{4y}  = 1)

3%9 = 3

So q = 3

p+q = 2+3 = 5

5%11 = 5

arnavghatage · Answer

I get confused between remainder and divisibility.

Take this question for  https://gmatclub.com/forum/what-is-the-smallest-positive-integer-to-plus-370089.html .

Could we have used remainder method there?

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