JEE Advanced 2019 · Paper 1 · Mathematics Q14

MathematicsNumerical+3Question 50 of 54

Let $AP(a; d)$ denote the set of all the terms of an infinite arithmetic progression with first term $a$ and common difference $d > 0$ . If

$AP(1;3)\cap AP(2;5)\cap AP(3;7) = AP(a;d)$

then $a + d$ equals ______

Your answer

Answer and worked solution

Official answer: 157

We need $x\equiv1\pmod3$ , $x\equiv2\pmod5$ and $x\equiv3\pmod7$ .

Solve. The numbers $\equiv3\pmod7$ that are $\equiv2\pmod5$ are $17,52,87,\dots$ . Among these, $52\equiv1\pmod3$ . So $x\equiv52\pmod{105}$ .

The intersection. It is $AP(52;105)$ , since 52 already exceeds every first term. So $a+d=52+105=157$ .

Answer: 157

Topics:Arithmetic progression Number systems

JEE Advanced 2019 · Paper 1, Mathematics question 14