CAT Number System Notes: Chapter-wise Weightage and Solved Questions

CAT Number System: 4 chapters, 0 practice questions and one solved question from each chapter.

Number System Chapter Matrix

Chapter	PYQs	Share of unit PYQs	Practice questions
Digit Problems, Floor Values and Number Construction	0	0%	0
Number Theory Word Problems and Diophantine Conditions	0	0%	0
Prime Factors, Divisors and Factorials	0	0%	0
Remainders, Divisibility and Modular Arithmetic	0	0%	0

One Solved Question from Each Number System Chapter

Question 1 · Quantitative Ability MCQ

Let $S (M)$ denote the sum of the decimal digits of a positive integer $M$ . Find

$S (k = 1 \sum 12 ⌊ \frac{k \cdot 1 0 ^{24}}{13} ⌋) .$

A.
198
B.
216
C.
225
D.
234

Correct Answer:

Step-by-Step Solution

Key idea: this is a complementary-pair floor sum. The denominator is $13$ and the numerator runs through $k = 1$ to $12$ , so pair $k$ with $13 - k$ . The huge power of $10$ is then handled by a borrow-chain digit sum.

Step 1: Let $N = 1 0^{24}$ . Since $13$ does not divide $N$ , none of $k N$ is divisible by $13$ for $k = 1, 2, \dots, 12$ .

Step 2: For a fixed $k \in {1, 2, \dots, 6}$ , write

$k N = 13 q + r,$

where $1 \leq r \leq 12$ .

Step 3: Then

$(13 - k) N = 13 N - k N = 13 N - (13 q + r) = 13 (N - q) - r .$

Rewrite the last expression as

$13 (N - q - 1) + (13 - r) .$

Since $13 - r$ is between $1$ and $12$ , we get

$⌊ \frac{( 13 - k ) N}{13} ⌋ = N - q - 1.$

Step 4: Therefore each complementary pair gives

$⌊ \frac{k N}{13} ⌋ + ⌊ \frac{( 13 - k ) N}{13} ⌋ = q + (N - q - 1) = N - 1.$

Step 5: There are $6$ pairs: $(1, 12), (2, 11), \dots, (6, 7)$ . Hence the whole sum is

$6 (N - 1) = 6 (1 0^{24} - 1) = 6 \cdot 1 0^{24} - 6.$

Step 6: Find the digit sum of $6 \cdot 1 0^{24} - 6$ . The number $6 \cdot 1 0^{24}$ is a $6$ followed by $24$ zeroes. Subtracting $6$ creates a borrow chain:

$6 \cdot 1 0^{24} - 6 = 5 23 nines 99 \dots 9 4.$

Its digit sum is

$5 + 23 \cdot 9 + 4 = 5 + 207 + 4 = 216.$

Common trap: replacing the sum of floors by the floor of the sum. Floors do not distribute over addition; the fractional parts are exactly what make the pairing work.

Answer: 216

chapter

CAT Number System Notes: Chapter-wise Weightage and Solved Questions

CAT Number System: 4 chapters, 0 practice questions and one solved question from each chapter.

A question from this chapter

Question 1

Let S (M) denote the sum of the decimal digits of a positive integer M . Find S (k = 1 \sum 12 ⌊ \frac{k \cdot 1 0 ^{24}}{13} ⌋) .

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