MBApyq

Six-digit number condition

CAT 2019 Slot 2 · Quantitative Ability · Medium · Number theory

This is a medium Quantitative Ability question from the CAT 2019 Slot 2 paper. It tests Number theory. The full answer key and a step-by-step explanation are below — try it yourself first, then reveal the solution.

All CAT 2019 Slot 2 questionsMore Number theory questionsAll CAT QA

In a six-digit number, the sixth, that is, the rightmost, digit is the sum of the first three digits, the fifth digit is the sum of first two digits, the third digit is equal to the first digit, the second digit is twice the first digit and the fourth digit is the sum of fifth and sixth digits. Then, the largest possible value of the fourth digit is

[TITA]

Answer

7

Explanation

Let the 6-digit number be d1d2d3d4d5d6d_1 d_2 d_3 d_4 d_5 d_6.

  • d2=2d1d_2 = 2 d_1
  • d3=d1d_3 = d_1
  • d5=d1+d2=d1+2d1=3d1d_5 = d_1 + d_2 = d_1 + 2d_1 = 3 d_1
  • d6=d1+d2+d3=d1+2d1+d1=4d1d_6 = d_1 + d_2 + d_3 = d_1 + 2d_1 + d_1 = 4 d_1
  • d4=d5+d6=3d1+4d1=7d1d_4 = d_5 + d_6 = 3 d_1 + 4 d_1 = 7 d_1

Since d4d_4 is a single digit (0≤d4≤90 \le d_4 \le 9) and d1≥1d_1 \ge 1: If d1=1  ⟹  d4=7d_1 = 1 \implies d_4 = 7. If d1≥2  ⟹  d4≥14d_1 \ge 2 \implies d_4 \ge 14 (not a single digit).

Thus, the largest possible value of d4d_4 is 7.

Related Number theory questions

Practise this under exam conditions

Sign in to solve it with a live timer, the on-screen CAT calculator, and streak and accuracy tracking across every question you attempt.

Solve in the workspace