# CAT 2019 Quant Question with Solution 67

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

1. 18
2. 19
3. 21
4. 20

Both the sequences are in arithmetic progression.

The common difference (${{d}_{1}}$ ) for the first sequence = 4

The common difference (${{d}_{2}}$ ) for the first sequence = 5

The first term common is 19.

The common terms will also be in arithmetic progression with common difference $LCM\left( {{d}_{1}},{{d}_{2}} \right)=LCM\left( 4,5 \right)=20$

Let there be ‘n’ terms in this sequence, then the last term would be ≤ 415

i.e. $a+\left( n-1 \right)d\le 415$

$\Rightarrow 19+\left( n-1 \right)\times 20\le 415$

$\Rightarrow \left( n-1 \right)\times 20\le 415-19$

$\Rightarrow \left( n-1 \right)\times 20\le 396$

$\Rightarrow \left( n-1 \right)=\left[ \frac{396}{20} \right]$ where [ ] is the greatest integer

$\Rightarrow \left( n-1 \right)=19$

$\Rightarrow n=20$

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