JEE Main 2025 Apr 4 Shift 2, Mathematics Q24: Number System And Logarithms

JEE Main2025Apr 4, Shift 2Mathematics

Let $m$ and $n$ , ( $m < n$ ) be two 2-digit numbers. Then the total numbers of pairs $(m, n)$ , such that $g cd (m, n) = 6$ , is ______

Your answer

Solution

Write $m = 6 a, n = 6 b$ with $a < b$ and $g cd (a, b) = 1$ . Since $10 \leq m, n \leq 99$ , we need $⌈ 10/6 ⌉ = 2 \leq a, b \leq 16 = ⌊ 99/6 ⌋$ .

So we must count pairs $(a, b)$ with $2 \leq a < b \leq 16$ and $g cd (a, b) = 1$ .

Going through each $a$ from 2 to 15 and listing $b$ 's in $(a, 16]$ that are coprime to $a$ :

$a = 2$ : $b \in {3, 5, 7, 9, 11, 13, 15}$ — 7 values.

$a = 3$ : $b \in {4, 5, 7, 8, 10, 11, 13, 14, 16}$ — 9 values.

$a = 4$ : $b \in {5, 7, 9, 11, 13, 15}$ — 6 values.

$a = 5$ : $b \in {6, 7, 8, 9, 11, 12, 13, 14, 16}$ — 9 values.

$a = 6$ : $b \in {7, 11, 13}$ — 3 values.

$a = 7$ : $b \in {8, 9, 10, 11, 12, 13, 15, 16}$ — 8 values.

$a = 8$ : $b \in {9, 11, 13, 15}$ — 4 values.

$a = 9$ : $b \in {10, 11, 13, 14, 16}$ — 5 values.

$a = 10$ : $b \in {11, 13}$ — 2 values.

$a = 11$ : $b \in {12, 13, 14, 15, 16}$ — 5 values.

$a = 12$ : $b \in {13}$ — 1 value.

$a = 13$ : $b \in {14, 15, 16}$ — 3 values.

$a = 14$ : $b \in {15}$ — 1 value.

$a = 15$ : $b \in {16}$ — 1 value.

Total $= 7 + 9 + 6 + 9 + 3 + 8 + 4 + 5 + 2 + 5 + 1 + 3 + 1 + 1 = 64$ .

Concept behind this question

Concept-wise practice with instant solutions on Fundamenthol.