Advanced Concepts in Numbers

These are special tools and observations that help you crack seemingly tough problems in seconds — especially useful in speed-based aptitude tests.

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Digital Root (also called Digit Sum)

Definition: Repeatedly add digits of a number until a single digit remains.

Example:

$$\text{Digital root of 9876} = 9 + 8 + 7 + 6 = 30 \Rightarrow 3 + 0 = 3$$

Why it's Useful

• Helps in checking divisibility by 9 and 3
• Used in verifying calculation errors
• Reduces big number calculations for options-based questions

Properties

• Digital root of a number is same as $\text{number} \mod 9$, except when divisible by 9 (result is 9, not 0)
• $\text{DR}(a + b) = \text{DR}(\text{DR}(a) + \text{DR}(b))$
• $\text{DR}(a \times b) = \text{DR}(\text{DR}(a) \times \text{DR}(b))$

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Cyclicity of Last Digits

Definition: Repeating pattern in the last digit of powers of numbers.

How to Use:
To find the last digit of $a^b$:

1. Find the cyclic pattern of base
2. Compute $b \mod \text{cycle length}$
3. Pick the corresponding digit in the cycle

Example: Last digit of $7^{222}$
Cycle for 7: 7, 9, 3, 1 → Length = 4
222 mod 4 = 2 → Answer = 9

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Remainder Theorems

1. Remainder when a number is divided by 9 or 3

Use digital root or sum of digits.

$$\text{Remainder of 7382 \div 9} = \text{Sum of digits} = 20 → 20 mod 9 = 2$$

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2. $a^b \mod m$

Use modular arithmetic + cyclicity (or Euler’s theorem for high level).

Example: $2^{100} \mod 5$
Cycle of $2^n \mod 5$: 2, 4, 3, 1
100 mod 4 = 0 → 4th element → Answer = 1

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3. Fermat's Little Theorem (Advanced)

If $p$ is a prime, then:

$$a^{p-1} \equiv 1 \mod p$$

Used when:

• $a$ is not divisible by $p$
• Needed to reduce large exponents

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Number of Zeroes in Factorial

Trailing Zeroes in $n!$

Count of zeroes = Number of times 5 is a factor (2 is always in excess):

$$\text{Zeroes in } n! = \left\lfloor \frac{n}{5} \right\rfloor + \left\lfloor \frac{n}{25} \right\rfloor + \left\lfloor \frac{n}{125} \right\rfloor + \dots$$

Example: Zeroes in 100!
= 20 + 4 = 24

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Factorial and Units Digit

To find the last non-zero digit of $n!$, eliminate 0s (remove 5s and matching 2s) and find the digit modulo 10. These are usually asked indirectly.

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Highest Power of a Prime in a Factorial

In $n!$, the exponent of prime $p$ is:

$$\left\lfloor \frac{n}{p} \right\rfloor + \left\lfloor \frac{n}{p^2} \right\rfloor + \left\lfloor \frac{n}{p^3} \right\rfloor + \dots$$

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Finding Number of Digits in a Number

To find number of digits in base 10:

$$\text{Digits in } n = \left\lfloor \log_{10}(n) \right\rfloor + 1$$

For $a^b$:

$$\text{Digits} = \left\lfloor b \log_{10} a \right\rfloor + 1$$

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Common Mistakes

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Example Problems

Q1. What is the digital root of $8532$?

8 + 5 + 3 + 2 = 18 → 1 + 8 = 9

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Q2. Find last digit of $3^{45}$

Cycle: 3, 9, 7, 1
45 mod 4 = 1 → Answer = 3

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Q3. Remainder when $23^456$ is divided by 5?

Cycle of 23 mod 5 = same as 3 mod 5 = 3 → Cycle: 3, 4, 2, 1
456 mod 4 = 0 → Answer = 1

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Q4. How many trailing zeroes in 200!?

$$\left\lfloor \frac{200}{5} \right\rfloor + \left\lfloor \frac{200}{25} \right\rfloor + \left\lfloor \frac{200}{125} \right\rfloor = 40 + 8 + 1 = **49**$$

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Visual Summary

→ Digital Root = mod 9 trick
→ Cyclicity = last digits of powers (mod 10)
→ Factorials = zeroes, prime powers
→ Mod Arithmetic = use for remainder-based exponent problems