CAT — Remainders
3 questions, free to view. Click any question to see the answer and explanation.
If 10⁶⁸ is divided by 13, the remainder is
5
8
9
4
9
Given:
Find the remainder when
10⁶⁸
is divided by 13.
Step 1: Express 10 modulo 13
10 ≡ −3 (mod 13)
Therefore,
10⁶⁸ ≡ (−3)⁶⁸ (mod 13)
Since 68 is even,
(−3)⁶⁸ = 3⁶⁸
Step 2: Find the repeating pattern
Using Fermat's Little Theorem,
3¹² ≡ 1 (mod 13)
Now,
68 = 12 × 5 + 8
Hence,
3⁶⁸
= (3¹²)⁵ × 3⁸
≡ 3⁸ (mod 13)
Step 3: Compute 3⁸ modulo 13
3² = 9
3⁴ = 9² = 81 ≡ 3 (mod 13)
3⁸ = (3⁴)²
≡ 3²
≡ 9 (mod 13)
Therefore,
10⁶⁸ ≡ 9 (mod 13)
Final Answer
The remainder is 9.
Answer:
C. 9
When 10¹⁰⁰ is divided by 7, the remainder is
3
4
1
6
4
Step 1: Reduce the base modulo 7
10 ≡ 3 (mod 7) → 10¹⁰⁰ ≡ 3¹⁰⁰ (mod 7)
Step 2: Find the pattern of powers of 3 modulo 7
3¹ ≡ 3, 3² ≡ 2, 3³ ≡ 6, 3⁴ ≡ 4, 3⁵ ≡ 5, 3⁶ ≡ 1 (mod 7)
The remainders repeat every 6 powers.
Step 3: Find the position of 100 in the cycle
100 ÷ 6 leaves remainder 4.
Therefore, 3¹⁰⁰ ≡ 3⁴ ≡ 4 (mod 7)
When 3³³³ is divided by 11, the remainder is
5
10
1
6
5
Given:
Find the remainder when
3³³³
is divided by 11.
Step 1: Find the repeating pattern
Since
3⁵ = 243
and
243 ÷ 11 leaves a remainder of 1,
we have
3⁵ ≡ 1 (mod 11)
Thus, the powers of 3 repeat every 5 terms modulo 11.
Step 2: Reduce the exponent
333 ÷ 5 leaves a remainder of 3.
Hence,
3³³³ ≡ 3³ (mod 11)
Step 3: Find the remainder
3³ = 27
27 ÷ 11 leaves a remainder of 5.
Therefore,
3³³³ leaves a remainder of 5 when divided by 11.
Answer:
A. 5
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