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CAT 2019 Slot 2 — QA Question 28

Arithmetic ProgressionEasy

The number of common terms in the two sequences: 15, 19, 23, 27, ...... , 415 and 14, 19, 24, 29, ...... , 464 is

Answer & solution

Correct answer: 20

  • 20

  • B

    18

  • C

    21

  • D

    19

Solution

Easy

Both lists are APs. Their common terms themselves form an AP whose common difference is the LCM of the two step sizes. Find the first common term, then count how many such terms fit before the earlier list runs out.

1

Identify the two APs.

Series 1: 15,19,23,…,415(common difference 4)Series 2: 14,19,24,…,464(common difference 5)\begin{aligned} &\text{Series 1: } 15,19,23,\dots,415 \quad\text{(common difference }4\text{)}\\ &\text{Series 2: } 14,19,24,\dots,464 \quad\text{(common difference }5\text{)} \end{aligned}
2

Common terms form an AP. First common term is 1919; the gap between common terms is LCM⁡(4,5)=20\operatorname{LCM}(4,5)=20.

Common terms: 19,39,59,…\begin{aligned} &\text{Common terms: } 19,39,59,\dots \end{aligned}
3

Bound by the shorter series. Series 1 stops at 415415, so common terms cannot exceed 415415. The largest valid one is 399399.

tk=19+(k−1)⋅20≤415⇒ 399=19+(k−1)⋅20(largest ≤415)⇒ k−1=19  ⇒  k=20\begin{aligned} &t_k = 19 + (k-1)\cdot 20 \le 415\\ &\Rightarrow\ 399 = 19 + (k-1)\cdot 20 \quad\text{(largest }\le 415\text{)}\\ &\Rightarrow\ k-1 = 19 \;\Rightarrow\; k = 20 \end{aligned}
Number of common terms=20\text{Number of common terms}=20

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CAT 2019 Slot 2 QA Q28: The number of common terms in the two sequences: 15, 19, 23, 27, ...... , 415 and 14, 19, 24, 29, ...... , 464 — Solution | TheCATExam