LUANDATO wrote:If p and q are positive integers, what is the remainder when 9^2p × 5^(p+q) + 11^q × 6^pq is divided by 10?
A)0
B)1
C)3
D)4
E)5
The OA is B.
Can any expert explain this PS question please? I don't have it clear. Thanks.
The remainder when a number is divided by 10 is its unit digit. For example, the remainder when 23
4 divided by 10 is 4.
We have 9^2p × 5^(p+q) + 11^q × 6^pq.
9^2p × 5^(p+q) + 11^q × 6^pq =
3^4p × 5^(p+q) + 11^q × 6^pq
Let's understand the deduction of the unit digit of exponents.
1. unit digit of the exponent of 3:
It follows a cycle of 4. Unit digit of 3^1 --> 3; 3^2 --> 9; 3^3 --> 7; 3^4 --> 1.
Thus, unit digit of 3^4p --> 3.
2. Unit digit of the exponent of 5:
Unit digit of the exponent of 5 is always 5.
3. Unit digit of the exponent of 6:
Unit digit of the exponent of 6 is always 6.
4. Unit digit of the exponent of 11:
Unit digit of the exponent of 11 is same as the unit digit of the exponent of 1, which is always 1.
Thus, unit digit of 3^4p × 5^(p+q) + 11^q × 6^pq --> 3 x 5 + 1 x 6 --> 5 + 6 --> 1.
The correct answer:
B
Hope this helps!
-Jay
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