If x is the remainder when 3 61284 is divided by 5 and y is the remainder when 4 96 is divided by 6, then what is the value of (2x – y)?
-2
Step 1 — Find x (remainder of \(3^{61284}\) by 5):
Powers of 3 mod 5 cycle with length 4: \(3,4,2,1\). Since \(61284=4\times 15321\), the exponent is a multiple of 4, so the remainder is the 4th term:
\[3^{61284}\equiv 3^{4}\equiv 1\pmod 5\;\Rightarrow\;x=1\]
Step 2 — Find y (remainder of \(4^{96}\) by 6):
For every \(n\ge 1\), \(4^{n}\equiv 4\pmod 6\) (since \(4\equiv 1\pmod 3\) makes \(4^{n}-4\) divisible by both 2 and 3). Hence \(y=4\).
Step 3 — Evaluate \(2x-y\):
\[2(1)-4=-2\]
Therefore \(2x-y=\mathbf{-2}\).
Similar Questions
If the 8-digit number 888x53y4 is divisible by 72, then what is the value of (7x + 2y), for the maximum value of y?
Which of the following is the smallest number that is a perfect square and is divisible by each of the numbers 6, 8 and 15?
If the seven-digit number 94x29y6 is divisible by 72, then what is the value of (2x + 3y) for x ≠ y ?
A four-digit pin, say abcd, of a lock has different non-zero digits. The digits satisfy b = 2a, c = 2b, d = 2c. The pin is divisible by ________.
If the 5-digit number 676xy is divisible by 3, 7 and 11, then what is the value of (3x - 5y)?
What is the remainder when we divide 5 70 + 7 70 by 74?
If the 5-digit number 688xy is divisible by 3, 7 and 11, then what is the value of (5x + 3y)?
If a number P is divisible by 2 and another number Q is divisible by 3, then which of the following is true?
A divisor is 15 times the quotient and 3 times the remainder. If the remainder is 40, find the dividend.
If each of the two numbers 5 16 and 5 25 are divided by 6, the remainders are R 1and R 2 respectively. What is the value of \(\frac{{{{\rm{R}}_{\rm{1}}}{\rm{ + }}{{\rm{R}}_{\rm{2}}}}}{{{{\rm{R}}_{\rm{2}}}}}?\)
Important Questions from Divisibility and Remainder
If the 8-digit number 888x53y4 is divisible by 72, then what is the value of (7x + 2y), for the maximum value of y?
If all positive divisors of 132 are arranged in descending order, then what digit will be at unit place of first divisor ?
If 3 2019 is divided by 10, then what is the remainder?
The number 3798125P369 is divisible by 7. What is the value of the digit P?
Consider all 3-digit numbers (without repetition of digits) obtained using three non-zero digits which are multiples of 3. Let S be their sum.
Which of the following is/are correct?
1. S is always divisible by 74.
2. S is always divisible by 9.
select the correct answer using the code given below:
