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What is the remainder, after division by 100, of 7^10 ? 15 Oct 2019, 21:26
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7 has a cyclicity of 4: 7-9-3-1 for its units digit.

The remainder for a power of 10 will be the last 2 digits of that number.

We don't really need to think much further as we are expecting a units digit of 9 (2 cycles 4 + 2 more spaces)... AC (d)
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What is the remainder, after division by 100, of 7^10 ? 16 Sep 2024, 06:58
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Bunuel can we directly go by units digit of 7 than doing it units digit of 49 thanks
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What is the remainder, after division by 100, of 7^10 ? 16 Sep 2024, 07:13
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MalachiKeti
Bunuel can we directly go by units digit of 7 than doing it units digit of 49 thanks
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Yes, you can.
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What is the remainder, after division by 100, of 7^10 ? 09 Nov 2024, 06:01
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kkalyan
What is the remainder, after division by 100, of 7^10 ?

(A) 1
(B) 7
(C) 43
(D) 49
(E) 70
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Here is a one minute video on how to solve it using Binomial Theorem:
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What is the remainder, after division by 100, of 7^10 ? 30 Sep 2025, 03:33
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Bunuel could you please clarify here in the answer choices if there would have been another choice with units digit 9, then what would have been the approach?
Bunuel


What is the remainder, after division by 100, of 7^10 ?

(A) 1
(B) 7
(C) 43
(D) 49
(E) 70

The remainder when 7^10 is divided by 100 will be the last two digits of 7^10 (for example 123 divided by 100 yields the remainder of 23, 345 divided by 100 yields the remainder of 45).

\(7^{10}=(7^2)^5=49^5\) --> the units digit of 49^5 will be 9 (the units digit of 9^even is 1 and the units digit of 9^odd is 9).

So, we have that \(7^{10}=49^5\) has the units digit of 9, thus the units digit of the remainder must also be 9. Only answer D fits.

Answer: D.
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What is the remainder, after division by 100, of 7^10 ? 30 Sep 2025, 07:13
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Aishna1034
Bunuel could you please clarify here in the answer choices if there would have been another choice with units digit 9, then what would have been the approach?

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Aishna1034 If I may help- If the choices had included, say, both \(49\) and \(29\) (or \(89\)), finding just the units digit wouldn't be sufficient.

The Complete Approach When Units Digit Isn't Enough

When multiple answer choices share the same units digit, you need to find the complete remainder when dividing by \(100\). Here's the systematic method:

Step 1: Calculate Powers Modulo 100
\(7^1 = 7\)
\(7^2 = 49\)
\(7^3 = 343\) → remainder \(= 43\) when divided by \(100\)
\(7^4 = 43 \times 7 = 301\) → remainder \(= 1\)
\(7^5 = 1 \times 7 = 7\) (pattern repeats!)

Step 2: Identify the Pattern
The remainders cycle every \(4\) powers: \(7, 49, 43, 1, 7, 49, 43, 1...\)

Step 3: Find Your Position in the Cycle
Since \(10 = 4 \times 2 + 2\), we know \(7^{10}\) has the same remainder as \(7^2 = 49\)

Process Diagnosis
The units digit approach (finding \(7^{10} \mod 10\)) only gives you the last digit. But remainder after division by \(100\) requires both the tens and units digits. This is why calculating the full pattern modulo \(100\) is essential when answer choices could share units digits.

Strategic Insight - Pattern Recognition Framework
This belongs to the "Remainder with Large Divisors" family of problems. When you see:
- Division by \(100\) → You need two-digit remainders
- Division by \(1000\) → You need three-digit remainders
- Multiple answer choices with same units digit → Calculate complete remainder, not just units

Decision Rule: If divisor > 10, always calculate the full remainder pattern, not just the units digit.

You can practice similar questions here (you'll find a lot of OG questions) - select Number Properties under Quant and choose Medium level questions focused on remainders and patterns.
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