Set A Solutions

A1. Answer: (b) 3,25,764 Check last two digits of each: 22 (not ÷4), 64 (÷4=16 ✓), 26 (not ÷4), 45 (not ÷4). Only (b) qualifies.

A2. Answer: (b) 78,904 Check last three digits: 326 (326÷8=40.75 ✗), 904 (904÷8=113 ✓), 412 (412÷8=51.5 ✗), 238 (238÷8=29.75 ✗). Only (b) works.

A3. Answer: (d) 7 Digit sum =6+3+x+2=11+x. This must be a multiple of 9. Since x is a single digit (0–9), the sum 11+x ranges from 11 to 20; the only multiple of 9 in this range is 18. So 11+x=18 ⇒ x=7. Check: number = 63,72; digit sum 6+3+7+2=18, divisible by 9. ✓

A4. Answer: (c) 2 Number: 5 x 4 8 3 2 — reading digit positions from the right: 2(pos1),3(pos2),8(pos3),4(pos4),x(pos5),5(pos6). Odd positions (1,3,5): 2+8+x=10+x. Even positions (2,4,6): 3+4+5=12. Difference =(10+x)-12 = x-2. For divisibility by 11, x-2=0 ⇒ x=2. Check: number = 524832; alternating sum =2+8+2-(3+4+5)=12-12=0 → divisible by 11. ✓

A5. Answer: (b) 4,326 Check even + digit-sum÷3: (a) 3542 even, sum=14 (not÷3) ✗; (b) 4326 even, sum=4+3+2+6=15 (÷3=5 ✓) → divisible by 6; (c) 5671 odd ✗; (d) 2213 odd ✗.

A6. Answer: (b) 1 Digit sum =8+4+2+9+1+7=31. 31 ÷ 3 = 10 remainder 1.

A7. Answer: (c) 5 Digit sum =2+7+x+4=13+x. Nearest multiple of 9 ≥13 is 18 → x=5.

A8. Answer: (c) 8 Number 9 2 x 3 8 9, positions from right: 9(1),8(2),3(3),x(4),2(5),9(6). Odd sum (1,3,5): 9+3+2=14. Even sum (2,4,6): 8+x+9=17+x. Difference =14-(17+x) = -3-x. For divisibility by 11: -3-x=0 ⇒ x=-3 (invalid), or -3-x=-11 ⇒ x=8. x=8. Matches option (c).

A9. Answer: (b) 5,432 Osculator test: last digit 2, double=4, remaining=543, 543-4=539; last digit 9, double=18, remaining=53, 53-18=35=7×5 → divisible. Check others quickly fail (5431, 5433, 5434 are not multiples of 7; 5432/7=776 exactly).

A10. Answer: (a) 1,001 1001 = 7×11×13, so it’s divisible by 13. Osculator check: last digit 1,×4=4,remaining=100,100+4=104=13×8 ✓. Others (1002,1003,1004) are not multiples of 13.

A11. Answer: (c) 12 ⌊50/5⌋=10, ⌊50/25⌋=2, ⌊50/125⌋=0. Total =10+2=12.

A12. Answer: (c) 18 ⌊20/2⌋=10, ⌊20/4⌋=5, ⌊20/8⌋=2, ⌊20/16⌋=1, ⌊20/32⌋=0. Total =10+5+2+1=18.

A13. Answer: (c) both x-y and x+y n=12 is even, so by Rule 1 (x-y always divides) and Rule 2 (even n → x+y also divides), both apply.

A14. Answer: (c) 6 Standard result: product of 3 consecutive integers is always divisible by 3! = 6.

A15. Answer: (c) 3 Digit sum =7+1+3+6+4=21. 21÷9 leaves remainder 3.

A16. Answer: (a) 4,275 Test each with divisibility by both 3 and 5: (a) 4275 → last digit 5 ✓, digit sum 4+2+7+5=18 (÷3 ✓) → divisible by 15; direct check 4275÷15=285 exact. (b) 3812 → last digit 2, fails the 5-test outright. (c) 5288 → last digit 8, fails the 5-test outright. (d) 7295 → last digit 5 ✓, but digit sum 7+2+9+5=23, not divisible by 3 → fails. Only (a) satisfies both conditions.

A17. Answer: (c) 4 Digit sum =3+x+7+4=14+x. This must be a multiple of 9; in the range 14 to 23 (as x runs 0–9), the only multiple of 9 is 18. So 14+x=18 ⇒ x=4. Check: number = 3,474; digit sum 3+4+7+4=18, divisible by 9. ✓

A18. Answer: (c) 8 Number 8,192 — digits from right: 2(1),9(2),1(3),8(4). Odd positions (1,3): 2+1=3. Even positions (2,4): 9+8=17. Alternating sum =3-17=-14. Since -14 = -11-3, this is 3 short of -11 (the nearest multiple of 11 below it, other than -22), so -14 ≡ -14+22 = 8 (mod 11) (adding 22, the next multiple of 11 that brings it to a value in 0–10). Remainder = 8 (direct check: 8192 = 11×744 + 8 ✓).

A19. Answer: (c) 9 ⌊40/5⌋=8, ⌊40/25⌋=1, ⌊40/125⌋=0. Total =8+1=9.

A20. Answer: (c) both x-y and x+y n=8 is even, so by Rule 1 (x-y always divides x^n-yn) and Rule 2 (even n means x+y also divides), both apply — same logic as A13, just with a different even exponent.